Math
Modeling
getting
started &
getting
solutions
K. M. Bliss
K. R. Fowler
B. J. Galluzzo
Publisher
Society for Industrial and Applied Mathematics (SIAM)
3600 Market Street, 6th Floor
Philadelphia, PA 19104-2688 USA
www.siam.org
funding provided by
The Moody’s Foundation in association with the Moody’s
Mega Math Challenge, the National Science Foundation
(NSF), and the Society for Industrial and Applied
Mathmatics (SIAM).
Authors
Karen M. Bliss
Department of Math & Computer Science,
Quinnipiac University, Hamden, CT
Kathleen R. Fowler
Department of Math & Computer Science,
Clarkson University, Potsdam, NY
Benjamin J. Galluzzo
Department of Mathematics,
Shippensburg University, Shippensburg, PA
design & Connections to
common core state standards
PlusUs
www.plusus.org
PRODUCTION
First Edition 2014
Printed and bound in the United States of America
No part of this guidebook may be reproduced or
stored in an online retrieval system or transmitted in
any form or by any means without the prior written
permission of the publisher. All rights reserved.
CONTENTS
1. introduction
2
2. defining the problem statement
10
3. making assumptions
15
4. defining variables
20
5. building solutions
25
6. analysis and model Assessment
32
7. putting it all together
40
appendices & reference
45
the world
around
us is filled
with
important,
unanswered
questions.
1. INTRODUCTION
The world around us is filled with important,
unanswered questions. What effect will rising sea
levels have on the coastal regions of the United States?
When will the world’s human population surpass
10 billion? How much will it cost to go to college in
10 years? Who will win the next U.S. Presidential
election? There are also other phenomena we wish to
understand better. Is it possible to study crimes and
identify a burglary pattern [1, 10]? What is the best
way to move through the rain and not get soaked
[7]? How feasible is invisibility cloaking technology
[6]? Can we design a brownie pan so the edges do not
burn but the center is cooked [2]? Possible answers to
these questions are being sought by researchers and
students alike. Will they be able to find the answers?
Maybe. The only thing one can say with certainty is
that any attempt to find a solution requires the use
of mathematics, most likely through the creation,
application, and refinement of mathematical models.
A mathematical model is a representation of a system
or scenario that is used to gain qualitative and/
or quantitative understanding of some real-world
problems and to predict future behavior. Models
are used in a variety of disciplines, such as biology,
engineering, computer science, psychology, sociology,
and marketing. Because models are abstractions of
reality, they can lead to scientific advances, provide the
foundation for new discoveries, and help leaders make
informed decisions.
This guide is intended for students, teachers, and
anyone who wants to learn how to model. The aim
of this guide is to demystify the process of how a
mathematical model can be built. Building a useful
math model does not necessarily require advanced
mathematics or significant expertise in any of the
fields listed above. It does require a willingness to do
some research, brainstorm, and jump right in and try
something that may be out of your comfort zone.
3
1: introduction
math modeling
vs. word problems
Modeling problems are entirely different than the types of word problems most of us encountered in math classes.
In order to understand the difference between math modeling and word problems, consider the following questions
about recycling.
1. The population of Yourtown is 20,000, and 35% of its citizens recycle their plastic water bottles. If each person uses
9 water bottles per week, how many bottles are recycled each week in Yourtown?
2. How much plastic is recycled in Yourtown?
The solution to the first question is straightforward:
0.35 × 20,000 people × 9
bottles
bottles
= 63,000
person × week
week
This type of question might appear in a math textbook
to reinforce the concept that we translate the phrase
“35% of” to the mathematical computation “0.35
times.” It is an example of what we would call a word
problem: the problem explicitly gives you all the
information you need. You need only determine the
appropriate math computation(s) in order to arrive at
the one correct answer. Word problems can be used
to help students understand why we might want to
study a particular mathematical concept and reinforce
important mathematical skills.
The second question is quite different. When you
read a question like this, you might be thinking, “I
don’t have enough information to answer this question,” and you’re right! This is exactly the key point: we
usually don’t have complete information when trying
to solve real-world problems. Indeed, such situations
demand that we use both mathematics and creativity.
When we encounter such situations where we have
incomplete information, we refer to the problem as
open-ended. It turns out that mathematical modeling
is perfect for open-ended problems. This question, for
example, might have been conceived because we saw
garbage cans overflowing with water and soda bottles
and then wondered how many bottles were actually
being thrown out and why they were not being
recycled. Modeling allows us to use mathematics to
analyze the situation and propose a solution to promote
recycling.
In the word problem example above, it is assumed
that each person in town uses 9 plastic water bottles per
week and that 35% of the 20,000 people recycle their
water bottles every time they use one. Are these reasonable assumptions? The number 20,000 is probably an
estimate of Yourtown’s population, but where is the
other information coming from? Is it likely that every
person in the town uses exactly 9 water bottles every
week? Is it likely that 35% of people recycle every water
4
the volume of plastic waste Yourtown sent to landfills
bottle they use while 65% of people never recycle any
last year,” then there is exactly one correct answer.
of their water bottles? Probably not, but maybe this is
However, it’s unlikely that you will ever have sufficient
an average value, based on other data. The first problem
information to find that answer. In light of this, you
doesn’t invite us to determine whether the scenario
will develop a model that best estimates the answer
is realistic; it is assumed that we will accept the given
given the available information. Since no one knows
information as true and make the appropriate
the true answer to the question, your model is at least
computations.
as important as the answer itself, as is your ability to
In order to answer the second (modeling) problem
explain your model.
above, you would need to research the situation for
In contrast to word problems, we often use the
yourself, making your own (reasonable) assumptions
phrase “a solution” (as opposed to “the solution”) when
and strategies for answering the question. The
we talk about modeling problems. This is because
question statement doesn’t provide specific details
people who look at the same modeling problem may
about Yourtown.
have different perspectives into its
You will have to determine
resolution and can certainly come up
what factors about Yourtown
people who look at the
with different, valid alternative solucontribute to the amount of plastic
same modeling problem
tions. It is worth noting that word
that gets recycled. It seems reasonmay have different
problems can actually be thought of
able to believe that the population
perspectives into its
as former modeling problems. That is
of Yourtown is an important factor,
resolution and can
to say, someone has already deterbut what else about the city affects
certainly come up
mined a simple model and provided
the recycling rate? The question
with different, valid
you with all the relevant pieces of
statement failed to mention what
alternative solutions.
information. This is very different
types of plastic you should be takfrom a modeling problem, in which
ing into account. It would be hard
you must decide what’s important and how to piece it
to quantify all plastic thrown away. Is it a reasonable
all together.
assumption to consider only the plastics from food and
Mathematical modeling questions allow you to
beverage containers if you believe those are the priresearch real-world problems, using your discoveries
mary plastic waste sources? You would have to do some
to create new knowledge. Your creativity and how you
research and make some assumptions in order to make
think about this problem are both highly valuable in
any progress on this problem.
finding a solution to a modeling question. This is part
If, after your research, you distill the original probof what makes modeling so interesting and fun!
lem into something very specific, such as “Determine
5
1: introduction
overview of the
modeling process
figure 1.
Real world problem
Building the model
defining
the
problem
Getting a
solution
repeat as
needed or as
time allows
research &
brainstorming
Making
assumptions
Defining
variables
Analysis & model
assessment
reporting results
This guide will help you understand each of the
components of math modeling. It’s important to remember
that this isn’t necessarily a sequential list of steps; math
modeling is an iterative process, and the key steps may be
revisited multiple times, as we show in Figure 1.
6
• Getting a Solution What can you learn from your
model? Does it answer the question you originally
asked? Determining a solution may involve penciland-paper calculations, evaluating a function, running
simulations, or solving an equation, depending on the
type of model you developed. It might be helpful to
use software or some other computational technology.
• Defining the Problem Statement Real-world
problems can be broad and complex. It’s important
to refine the conceptual idea into a concise problem
statement which will indicate exactly what the
output of your model will be.
• Making Assumptions Early in your work, it may
seem that a problem is too complex to make any
progress. That is why it is necessary to make assumptions to help simplify the problem and sharpen the
focus. During this process you reduce the number of
factors affecting your model, thereby deciding which
factors are most important.
• Analysis and Model Assessment In the end, one must
step back and analyze the results to assess the quality
of the model. What are the strengths and weaknesses
of the model? Are there certain situations when the
model doesn’t work? How sensitive is the model if you
alter the assumptions or change model parameters
values? Is it possible to make (or at least point out)
possible improvements?
• Defining Variables What are the primary factors influencing the phenomenon you are trying to
understand? Can you list those factors as quantifiable variables with specified units? You may need to
distinguish between independent variables, dependent
variables, and model parameters. In understanding these ideas better, you will be able both to define
model inputs and to create mathematical relationships, which ultimately establish the model itself.
• Reporting the Results Your model might be awesome, but no one will ever know unless you are able to
explain how to use or implement it. You may be asked
to provide unbiased results or to be an advocate for a
particular stakeholder, so pay attention to your point
of view. Include your results in a summary or abstract
at the beginning of your report.
We will address the components in more detail one by one, but we note again that this should not be thought of as a
checklist for modeling. Throughout the process of building your model, you’ll likely move back and forth among the
components. Take careful notes as you go; it’s easy to get caught up in the modeling process and forget what you’ve
done along the way!
7
1: introduction
primary examples used
throughout this guide
We demonstrate the modeling process by looking at three
modeling questions in detail. We state those problems
directly below and then explore them throughout the
remainder of this guide.
Plastics aren’t the only problem. So many of the
materials we dispose of can be recycled. Develop a
mathematical model that a city can use to determine
which recycling method it should adopt. You may
consider, but are not limited to:
waste not, want not: putting
recyclables in their place
(A selection from Moody’s Mega Math Challenge:
2013 Problem. The full question and a solution paper
submitted by Team 1356 from Montgomery Blair High
School, Silver Spring, Maryland, coached by David Stein
and with student members Alexander Bourzutschky,
Alan Du, Tatyana Gubin, Lisha Ruan, and Audrey Shi, is
included as Appendix B.)
Plastics are embedded in a myriad of modern-day
products, from pens, cell phones, and storage containers
to car parts, artificial limbs, and medical instruments;
unfortunately, there are long-term costs associated with
these advances. Plastics do not biodegrade easily. There
is a region of the Northern Pacific Ocean, estimated
to be roughly the size of Texas, where plastics collect
to form an island and cause serious environmental
impact. While this is an international problem, in
the U.S. we also worry about plastics that end up in
landfills and may stay there for hundreds of years. To
gain some perspective on the severity of the problem,
the first plastic bottle was introduced in 1975 and now,
according to some sources, roughly 50 million plastic
water bottles end up in U.S. landfills every day.
• Providing locations where one can drop off pre-sorted
recyclables
• Providing single-stream curbside recycling
• Providing single-stream curbside recycling in addition
to having residents pay for each container of garbage
collected
Your model should be developed independent of current
recycling practices in the city and should include
some information about the city of interest and some
information about the recycling method. Demonstrate
how your model works by applying it to each of the
following cities: Fargo, North Dakota; Price, Utah;
Wichita, Kansas.
8
Outbreak? Epidemic? Pandemic?
Panic?
We all dread getting sick. Years ago, illness didn’t
spread very quickly because travel was difficult and
expensive. Now thousands of people travel via trains
and planes across the globe for work and vacation every
day. Illnesses that were once confined to small regions
of the world can now spread quickly as a result of one
infected individual who travels internationally. The
National Institutes of Health and the Centers for Disease
Control and Prevention are interested in knowing
how significant the outbreak of illnesses will be in the
coming year in the U.S.
Will it Thrill Me?
Amusement parks are typically open during the summer
months, when the heat and humidity are almost
unbearable. The lines for the most popular rides can
sometimes be hours long, leaving you to decide whether
you should spend your limited time at the park waiting
to ride the newest, most popular roller coaster (with
the longest line) or instead riding several, possibly less
exciting, roller coasters.
Unfortunately there is no real metric for scoring
roller coasters, although an extensive database exists
with information about many rides (see rcdb.com).
Innovative roller coaster engineers certainly set out to
design a thrilling roller coaster, but what makes a roller
coaster exciting and fun? Create a mathematical model
that ranks roller coasters according to a thrill factor that
you define.
9
2. defining the
problem
statement
Modeling problems are often open ended. Some math
modeling problems are clearly defined, while others
are ambiguous. This means there is an opportunity for
creative problem solving and interpretation. In some
cases, it is up to the modeler to define the outputs of
the model and which key concepts will be quantified.
Defining the problem statement requires some research
and brainstorming. The goal is a concise statement that
explains what the model will predict.
To see how a math modeling question can be
interpreted in different ways, consider the roller coaster
problem proposed earlier: rank roller coasters according
to how thrilling they are. The word “thrilling” here
is open to several interpretations. There are many
reasonable possibilities in defining and quantifying
“thrilling.”
For example, one student’s definition of a thrilling
ride may be a combination of the maximum height
and the number of loops, while another student values
a combination of length of a ride and the maximum
speed. If these individuals ranked the same list of roller
coasters, their ranking systems would likely produce
different results, neither of which would be “the”
correct ranking. The modeler has room to be creative in
deciding how to define “thrilling” but must make sure
that no matter what definition she decides upon, there is
a systematic ranking that incorporates quantifiable (i.e.,
measurable) aspects of a roller coaster.
Perhaps you’re thinking that the reason the students
above didn’t come up with “the” one correct ranking
with either of the previous models is because neither
of those models incorporate sufficiently sophisticated
mathematics. Suppose that we can leverage tools from
mathematics and physics to help answer this question.
Given the design of a particular roller coaster, we might
compute, among other things, velocities and g-forces a
rider would experience. Even with this information in
hand, it’s not obvious how to use that information to
rank roller coasters.
Consider four different roller coasters (A, B, C,
and D). Coaster A has a larger maximum velocity than
B, but B has a higher average velocity. Which is more
thrilling? How would these two rank against roller
coaster C, which attains a g-force twice as large as A’s
or B’s but only does so for 10 seconds of the entire
ride? Suppose that roller coaster D never reaches that
g-force but sustains g-forces only .5 g less for more
than 50 seconds. Which is more thrilling? The modeler
must choose a definition for thrilling. Eventually, when
communicating the results, a modeler will need to
explain why decisions were made and will discuss the
strengths and weakness of the model.
In the previous discussion we mentioned just
a few measurable aspects of roller coasters that one
could use to define “thrilling,” including maximum
height, the number of peaks, the maximum velocity,
or some combination of these. Where does one get a
list like this? They come from a process we refer to as
brainstorming. Brainstorming is part of the problemsolving process where spontaneous ideas are allowed to
flow without evaluation and interruption.
10
Figure 2
Example of mind map to explore
the definition of “best”
most
participation
least overall
cost to city
“best”
recycling
method
processes the
most recyclables
The roller coaster example demonstrates that
brainstorming at the beginning of a project is an
essential process that helps reveal different directions
that the math model can take. A brainstorming session
may include listing all of the things that make a roller
coaster thrilling and then digging deeper to see how
those properties are measured. At the beginning of the
process, however, one may want to just let the ideas
flow and then prune the list later after determining what
resources are available. This process is related to making
assumptions, which we will talk about in more detail
in the next section.
We’ll look at the brainstorming process in detail
by showing how it might work within the context of
the recycling problem. In this problem, we want to
determine which recycling method would be best for
a city to adopt. The word “best” needs to be clearly
defined, and there are multiple ways to do that. Let’s
imagine that we are on a team that works together to
discuss this, and we think of three possible ways to
define “best” in this problem.
In order to organize our thoughts, we might
use a mind map, as in Figure 2, to give us a visual
representation of our initial round of brainstorming.
A mind map is a tool to visually outline and organize
ideas. Typically a key idea is the center of a mind map
and associated ideas are added to create a diagram
that shows the flow of ideas. In Figure 2, we focus on
the definition of “best,” with three possible definitions
branching off to be further explored. From here, we
can focus our attention on one of the three branches
at a time. Let’s think about the least-cost option first.
We probably can’t determine how much any recycling
program costs without knowing more about the
recycling program, so a good place to start is to ask the
question “What kinds of recycling programs exist?”
If we aren’t familiar with different types of recycling,
we might need to do some research to see what kinds
of programs exist.
If you are working on a long-term modeling
project and you have lots of time, you’ll want to do an
extensive search to find learn everything you can about
the problem. You’ll also want to find out if others have
considered modeling this situation. If you are working
on a problem and you have a fairly short time frame,
you’ll need to be careful to not spend all of your time
on the internet researching the problem. Instead, do
a quick, preliminary internet search to get a broad
perspective (without getting too far into the “weeds”).
Suppose that the list of recycling methods consists
of drop-off center, curbside single-stream, curbside
(presorted), and pay-as-you-throw. Next, we need to
consider the costs. Let’s focus on one of the branches,
say single-stream curbside pick-up of recyclables. We
then ask ourselves, “What contributes to cost for this
method?” Then we ask, “For each of those costs, what
is the dependence on the properties of the city?”
11
2: defining the problem statement
Figure 3
Possible mind map under the assumption
that “best” means least cost
A possible final mind map for the least-cost
approach is shown in Figure 3.
Although we will not include the details
here, you can imagine that we could proceed
in a similar fashion for each of the three
definitions of “best.” We would then choose
one of the three possibilities, define the
problem statement in terms of this choice, and
move forward from there to develop a model.
During the brainstorming process, explore the
problem from different perspectives as if you
had access to all the data you could ever need.
In the next section, Making Assumptions, we’ll
discuss exactly what you can do if you can’t
find all of the data you need. Don’t discount
any idea simply because you don’t think you’ll
be able to find sufficient data.
One of the most important aspects of
brainstorming is to let the ideas flow freely,
especially if done in a group. It is best at this
initial phase to stay positive and be openminded. This part of the modeling process is
about creativity, so it is important that there is
no criticism of anyone’s ideas or suggestions.
What seems like a ridiculous approach may
later seem innovative after some more thought,
so make note of everything! Also, even if your
idea isn’t perfect, it might inspire someone else
to come up with an even better suggestion.
After you’ve explored the problem and
considered several possible approaches,
you can step back and look at the possible ways
a model might be constructed. Your intuition
will help you analyze your brainstorming
results and decide on a reasonable
problem statement.
how many are ne
dropoff
center
processes
the most
recyclables
operational cos
likelihood of pa
incentives/refu
curbside
single
stream
operational cost
likelihood of par
efficiency
“best”
recycling
method
least
overall
cost to
city
curbside
(presorted)
operational cost
likelihood of
participation
most
participation
pay as
you
throw
12
operational cost
likelihood of par
any are needed?
tional cost
hood of participaton
ives/refunds?
onal cost
ood of participation
size of city
how many/square mile?
population
how much waste can the center process?
start-up? (fixed cost)
processing costs/recyclable
distance to center
how far are people willing to drive?
probability based on data?
likelihood to participate
cost-benefit analysis
Limited scope (beverage containers only)
start-up? (fixed cost)
start-up? (fixed cost)
processing center
processing costs/recyclable
trucks
start-up?
(fixed cost)
how many
are needed?
GAS
area of city
onal cost
ood of
pation
onal cost
area of
city
population
start-up?
(see above
map)
truck capacity
maintenance
processing
center
probability
based on data?
number of trucks
number of trucks
employees
ood of participation
mileage year
how many?
single stream or pre-sort mapping
probability based on data?
13
wage
truck
capacity
2: defining the problem statement
in summary
1
Often math modeling questions are worded in ways that allow for multiple approaches, so you
should develop a concise restatement of the question at hand.
2
3
Focus on subjective words that can be interpreted in different ways. Also, identify words that
are not easily quantified. Examples include best, thrilling, efficient, robust and optimal.
Explore the problem by doing a combination of research and brainstorming, keeping in mind
your time constraints.
4
Keep an open mind and a positive attitude; you can prune out ideas later that are not realistic.
5
Brainstorming should be approached as if you had access to any data you need.
6
7
Visual diagrams, such as mind maps, can be a powerful tool leading to the structure of the
model. Consider using the website freemind (http://freemind.sourceforge.net/wiki/index.php/Main_Page) [5].
In the end, you should have a concise statement that explains what the model will measure
or predict.
Activity
Create a mind map for the disease-spreading problem.
14
3. Making
Assumptions
In presenting any scientific work to others, you
need to explain how the results were achieved with
explicit details so that they can be repeated. If you are
explaining a chemistry experiment, for example, you
need to list (among other things) which chemicals
were used in what quantities and in what order. Other
chemists would expect similar results only when they
used the same chemicals and procedure.
The list of assumptions for a mathematical model are
as critical as the experimental procedure in performing
a chemistry experiment. The assumptions tell the reader
under what conditions the model is valid. Making
assumptions can be one of the most intimidating parts
of the modeling process for a novice, but it need not
be! Assumptions are necessary and help you make a
seemingly impossible question much more tractable.
Many assumptions will follow quite naturally from
the brainstorming process. For the recycling problem,
some of our assumptions follow directly from the
questions we asked during the brainstorming session,
as on the following page.
Let’s further examine the assumption about how
many people would make use of drop-off centers
(termed “likelihood of participation”). The two extremes
would be to assume that the 100% of the people near a
recycling center would use it or that none would use it.
Neither of these seems like a reasonable assumption, so
what would be a better assumption? The students whose
solution to this problem appears in Appendix B decided
they would do some investigation and see if there has
been any successful research on participation rates in
drop-off centers. They found a study that had been
done in Ohio that estimated about 15% of households
participated in drop-off center recycling, and made an
assumption that this rate would hold in every city across
the U.S.
One might ask if it is safe to assume that across
the U.S. 15% of households will participate in drop-off
center recycling if it is available. Is it true that residents
of Arizona will behave the same way residents of Ohio
do? Certainly some cities would garner a participation
rate much higher than 15%, while other cities would
have a significantly lower participation rate. In fact,
what are the chances that any city would actually have
a participation rate of exactly 15%? In some sense, one
might say that assigning one participation rate to every
city across the U.S. is a ridiculous assumption.
In response to that line of thinking, remember
two things. First, remember that one must make
assumptions in order to make a model. It is not
practical or feasible to poll every citizen of every city
to determine who will bring recyclables to a dropoff center. If we had to rely on data with that level of
certainty at every juncture of the modeling process,
we would never get any work done. It’s practical and
important to make reasonable assumptions when we
cannot find data.
15
3: mAKING ASSUMPTIONS
Brainstorming Question
ASSUMPTION
brainstorming question
The best recycling method
What is meant by the “best”
will be interpreted to mean
recycling method?
the least cost to
the city.
We consider only four
recycling programs:
What recycling methods should
drop-off centers, single-stream
we consider?
curbside, presorted curbside, and
pay-as-you-throw.
What contributes
to cost for the
drop-off center method?
The cost of drop-off centers depends only
on the number of drop-off centers, the
quantity of recyclables that pass through
each center, and the costs to operate
each center.
The number of drop-off centers
What is the dependence
on the properties
of the city?
needed depends on the area of the
city, the population of the city, and
the likelihood of participation.
16
Second, you are developing a model that is intended
to help one understand some complex behavior or assist
in making a complex decision. It is not likely to predict
the exact outcome of a situation, only to help provide
insight and predict likely outcomes. When you provide
a list of your assumptions, you’ve done your part to
inform anyone who might use your model. They can
decide whether they think your assumption is or is not
appropriate to model the behavior they are interested
in predicting. In the Analysis and Model Assessment
section, we’ll discuss in more detail ways in which you
can examine some of the impacts of your assumptions.
It’s entirely possible that you may search and search
and never find the data you need to make an “educated”
guess about a parameter in your model. That’s fine;
simply make a note in your write-up that future work
might include further investigation in that area. If Team
1356 had not found any estimates for recycling rates,
they might have assumed that the recycling rate was
50% in the absence of other data (since it’s the mean of
the two extreme cases). That would have been a better
assumption than either of the extremes (all residents
recycle or no residents recycle). They also might have
determined that 25% seemed reasonable (based on their
own experiences or intuition) and moved forward with
that number. All of these are appropriate as long as they
are included as assumptions.
Choice of assumptions may also be dictated by the
mathematical tools available. Both the National Institute
of Health and the Centers for Disease Control and
Prevention use mathematical modeling to help them
understand the spread of infectious diseases. While their
models may be quite sophisticated, they are actually
built upon many of the simple principles we will discuss
here, which evolve from relatively few assumptions. Let’s
focus on determining the number of people who have
the disease over time by considering models at multiple
math levels.
One of the simpler models for disease propagation
can be created if we assume that the disease spreads at a
constant rate. For example, we might assume that each
person who has the disease will spread the disease to 3
people per day or that each person spreads the disease to
just 1 person every 5 days. As we move forward, we will
refer to this as the constant-rate disease model.
Transmission rate drives the spread of disease, and
the assumption that it remains constant over time seems
unlikely for the duration of the disease. If we have
knowledge of calculus and differential equations, we
can arrive at another model that accounts for varying
transmission rate.
17
3: mAKING ASSUMPTIONS
it is legitimate to make a reasonable assumption,
In order to decide how the transmission rate should
determine how it affects the model moving forward,
change with time, it might be helpful to think about
and make adjustments to improve the outcome. You
the mechanism behind disease transmission: infected
can see an example of this later in the Analysis and
people somehow in contact with susceptible individuals.
Model Assessment section. Make a careful list all of the
It makes sense, then, to believe that the transmission
assumptions you make along the way;
rate depends on how many people
a good modeling paper includes a
are infected and how many
how does one know
list of assumptions in the write-up.
people are susceptible. We might
which is the “best”
Additionally, keep track of all the
assume that the transmission rate
assumption to make?
resources used so that you can create a
is directly proportional to the
there is no easy answer
bibliography.
product of the number of people
to this question; But
With all of these options, how
infected and the number of people
be sure to acknowledge
does one know which is the best
who are susceptible. We will refer
the assumptions you’ve
assumption to make? There is no easy
to this model as the varying-rate
made and discuss their
answer to this question; the most
disease model. We will revisit both
limitations.
important thing is to acknowledge the
disease models in later sections.
Some assumptions are made
assumptions you’ve made and, when
at the beginning of the modeling
appropriate, discuss the limitations
process, while others are made as you proceed through
that might arise from your assumptions.
the modeling process. The modeling process is iterative;
18
in summary
1
Activity
Assumptions often come naturally
Build on the brainstorming from the
from the process of brainstorming and
previous section about the roller coaster
defining the problem statement.
model. Certainly we did not uncover all
the ways in which roller coasters are
2
You should do some preliminary research
considered thrilling. Define the problem
and may find data to help you make
statement in your own words, based on
assumptions. In the absence of relevant
your understanding of the problem.
data, make a reasonable assumption and
Finally, take your work one step further
justify the assumption in your write-up.
and list the assumptions on which you
could build your model.
3
Different assumptions can lead to
different, equally valid models at
different mathematical levels.
4
Not all assumptions are made during the
initial brainstorming. Some come as you
move through the modeling process. Keep
track of the assumptions you make and
include a list of assumptions in your
write-up of the model.
19
4. defining
variables
another term
for output.
With the problem statement clearly defined and an
initial set of assumptions made (a list that will likely
get longer), you are ready to start to define the details
of your model. Now is the time to pause to ask what
is important that you can measure. Identifying these
notions as variables, with units and some sense of their
range, is key to building the model.
The purpose of a model is to predict or quantify
something of interest. We refer to these predictions
as the outputs of the model. Another term we use
for outputs is dependent variables. We will also have
independent variables, or inputs to the model. Some
quantities in a model might be held constant, in which
case they are referred to as model parameters. Let’s look
at a few simple examples that will help you distinguish
between these concepts. We’ll also see how they depend
on your viewpoint and the problem statement.
the inputs of
the model.
The purpose of a model is to predict or
quantify something of interest. We refer to
these as the outputs of the model.
20
example 1:
Painting A House
Suppose that we plan to paint a house. We’ll need to know the dimensions of the house so that we can find the
surface area, SA (ft2). We also need to know the efficiency, E (ft2/gal), of the paint, which tells us how many square feet
a gallon of the paint can cover. Keep in mind that the efficiency varies from brand to brand. We let V (gal) be the volume of paint in gallons that we need. Here, knowing the units for efficiency can help reveal the relationship between
the variables:
E=
surface area
SA
=
volume
V
Notice that we can rewrite this
relationship and use any of the following
three equivalent relationships:
1
E = SA / V
2
SA = E · V
3
V = SA / E
dependent variable, and E is a constant model
Whether something is a dependent or independent
parameter.
variable or a parameter often relies on the perspective
Suppose instead that you are a homeowner and
of the modeler and the problem statement. Imagine
want to choose from among five
that you own a painting company
and always use CoversItAll brand
brands of paint to buy to minimize
whether something
paint. When a client hires you, you
the amount necessary to paint your
is a dependent or
take measurements of the house,
house. Under this scenario, the
independent variable
and then you want to know how
surface area of your house is a conor a parameter
much paint you’ll need to complete
stant, so this is treated like a model
often comes from
the job. In this scenario, the the
parameter. If we know the efficiency
the perspective of
of each of the five brands of paint,
efficiency of CoversItAll paint is
the modeler and the
we would again use equation (3), but
constant, so that is a model paramproblem statement.
eter. You would use equation (3),
this time with E as the input variable
with the surface area of the home as
and volume as the output. Thus,
the input and the volume of paint needed as an output.
in this case, E is the independent variable and V is the
Therefore, SA is the independent variable, V is the
dependent variable.
21
example 2:
Constant-Rate Disease Model
For another example, recall the constant-rate disease
model discussed in the previous section, wherein we
want to determine the number of infected individuals
at any given time. Using this problem statement, let t
denote the time in days. This will be the input (independent variable). Let I(t), the number of infected individuals at time t, be the output of the model (the dependent
variable). In this constant-rate disease model, we assume
that each person who has the disease will spread the
disease to a certain number of people in a given fixed
time period. We define the parameter τ to be the time
period during which each infected person will transmit
the disease to r other people (so r is also a parameter).
Further, we might define a parameter I0 to be the initial
number of infected individuals. In other words, I(0) =
I0. For example, if we have I0 = 10, τ = 2 days, and r = 3,
then we are considering a population starting with 10
infected individuals, where each infected person transmits the disease to 3 uninfected individuals every 2 days.
These simple examples show that the problem
statement will guide what the dependent variable (i.e.,
your model output) is going to be. Dependent variables
and parameters will often be determined by both your
assumptions and the availability of information. The key
idea is that independent variables cause a change in the
dependent variables. Let’s look at a more complex model
that demonstrates how submodels may be needed to
provide input to the overarching model.
The key idea is that
independent variables
cause a change in the
dependent variables.
22
example 3:
determining recycling costs
For the recycling problem, we seek a model that
will predict the cost for a city to implement and run
a recycling program. Hence, the output of the model
should be in dollars. What will the inputs to the model
be? We can use the results of our brainstorming session
to help us. Let’s consider the model to determine the
cost of a drop-off center. Based on the brainstorming
shown in Figure 3, the cost of using drop-off centers
in a given city could depend on how many are needed,
the operational cost, the likelihood that people will
participate, and the possibility of refunds or incentives.
Let’s consider the approach taken in the solution
provided in this guide. These students calculated
the cost of a drop-off center, based on the cost to
maintain the center, as well as the revenue the center
would create. This latter fact depends directly on the
participation rate by the local population. So, the
probability of participation by a single household
was needed first. Determining the single-household
probability rate is an example of how a submodel can be
used to generate input to the main model.
Continuing this line of thought, the students based
the likelihood that a household would participate on its
distance from a drop-off center. Thus, the distribution
of houses throughout the city and the location of
drop-off centers must be understood. This team
assumed that each city was a square and that houses
were aligned on grids. To determine the placement of
the drop-off centers within the city grid (which will
actually determine how many will be needed), students
calculated a maximum distance, d, that citizens would
be willing to drive per week to a drop-off center. Using
that distance, drop-off centers were placed on the grid
so that they did not overlap yet the entire city would be
covered.
Thus, for this approach, note that d is an input to
determine where to place the drop off-centers, yet d
itself needs to be determined first (because it certainly
isn’t clear what a reasonable value of d might be nor
is there one known value that suits everyone in the
United States). Therefore, ultimately we can use a math
modeling approach to determine d based on some
assumptions as well. This team decided that d depends
on the cost to get to a drop off-center, and this would
depend on the price of gas, gas mileage for a typical
car, and the amount that a household would be willing
to pay to recycle per month. Values for these model
parameters were found in the literature through some
digging into available resources.
To summarize their approach, they assumed that
the following were model parameters to find d:
• People would be willing to pay $2.29 to recycle per
month or $0.53 per week.
• People would make biweekly trips to the center.
• The average price of a gallon of gas is $3.784.
• Th
e average mileage of a passenger car is 23.8 miles/gallon.
23
4: defining variables
($0.53/week) · (23 miles/gallon)
d = $3.784/gallon
With a value of d in place, students were able to take
into consideration the size of the city grid and then
partition the city to place the appropriate number of
recycling drop-off centers to cover the entire city. Next,
the team wanted to propose a method for predicting
the likelihood of participation, but this couldn’t be done
until the previous submodels were developed.
Note that the inputs, or independent variables,
for their cost model for drop-off centers were the area
of the city, the population, the average number of
people in a household, the maximum distance citizens
would be willing to drive, and the number of drop-off
centers needed. However, more modeling was needed
to determine values for many of these inputs, as we
demonstrated with the input, d. Notice that these are
specific details that are implied from the brainstorming
shown in the mind map in Figure 2 but at this phase in
the process, more detail (and additional brainstorming
and assumptions) is needed.
/2 = 1.66 miles/week.
in summary
1
The problem statement should determine
the output of the model. The output
variables themselves will be dependent
variables.
2
The results of the initial brainstorming
can provide insight into which variables
should be independent variables and
which should be fixed model parameters.
3
4
Keep track of units because they can
reveal relationships between variables.
You will likely need to do some research
and make additional assumptions to obtain
values for necessary model parameters.
5
Submodels or multiple models may be
needed to generate some of the model
input.
Activity
Determine the dependent and independent
variables for ranking roller coasters based on
how thrilling they are. What are some possible
model parameters?
24
5. Building
Solutions
Now that you have an initial mathematical model,
you will need to use that model to generate preliminary
answers to the question at hand. The approach you take
of course depends on the type of model you have and
your background in mathematics. It may involve simply
considering some different values of certain parameters
to see how the output changes, it may involve
techniques from calculus or differential equations, or
it may involve using graphs to understand trends in
data. In this chapter we will give you some strategies for
choosing how to solve your problem.
When you first approach any mathematical
problem, you often look into your personal tool kit
for a mathematical technique to use. Sometimes, if we
start with the incorrect approach, a better approach
will naturally emerge. So, the important thing is to just
tackle it and see what happens! The following questions
may help you.
• Have I seen this type of problem before?
•I
f so, how did I solve it? If not, how is this
problem different?
•D
o I have a single unknown, or is this a multivariable problem with many interdependent
variables?
• Is the model linear or nonlinear?
•A
m I solving a system of equations
simultaneously, or can I solve sequentially?
•W
hat software or computational tools are
available to me?
•W
ould a graph or other visual schematic help
provide insight?
•C
ould I approximate my complicated model with
a simpler one?
•C
an I hold some values constant and allow
others to vary to see what is going on?
25
5: building solutions
Certainly there are times when it is clear what mathematical technique is required (for example, factoring, finding
zeros of a polynomial or a function, integrating a function, simulating a model over time to understand how output
evolves, etc.). Other times, when it is not clear how to proceed, it may be helpful to analyze simple examples,
special cases, or related problems. Even a “guess-and-check” approach can sometimes provide some deep insight.
Mathematical experiments may be facilitated with a graphing calculator or computational software such as Excel,
Mathematica, or Maple.
Multiple approaches can be taken to build a solution. We will show several approaches for each of the following
examples, which appear in order of increasing mathematical level. We also show how to leverage software to assist
you in finding solutions.
Example 1:
Acid Rain
Let’s consider approaching a general problem by building different models that may lead to different results. Suppose
we want to determine how acid rain is impacting the water resources in your town. We already know from our
previous section that this open-ended statement needs to be refined into a concise problem statement. Let’s suppose
we developed the following problem statement after some brainstorming and mind-mapping: Measure the levels of
sulfur dioxide (SO2) and nitrogen monoxide (NO) at four different locations and use the results to determine the best
place for water to be extracted.
Approach 1.
Approach 2.
Approach 3.
Ranking Given the measurements
for each location, rank the
locations as safest, second safest,
third safest, and least safe. This
allows a qualitative approach
but incorporates at least some
quantitative analysis, since it
requires us to assess the importance
of SO2 vs. NO and define the
term safe.
Equation-Based Solution We can
assess the difference in importance
between SO2 and NO, and create
an equation that assigns a score
to each location. The location
with the highest score wins. This
requires algebraic modeling and
manipulation as well as ideas
of proportionality.
Qualitative Comparison We can
decide if any of the sites is too
polluted. For example, if one site
has the highest levels of both SO2
and NO, then that site should not
be used. This reduces the problem
to choosing among three sites, and
the same ideas can be applied to
reduce them to two and then to a
final one.
26
Example 2:
Constant-Rate Disease Model
Often, when modeling real-world phenomena, we are interested in forecasting future values. That is, we want to
explore how the value of something we care about changes over time. We will discuss several approaches that we
could take to find a solution to the constant-rate disease model (in which we want to determine the number of people
who are infected with a disease under the assumption that each person afflicted with a disease spreads it to exactly r
people over a given time period τ). For this example, we’ll let I0 = 10, τ = 2 days, and r = 5. In other words, we have
a situation in which 10 people were infected with a disease that they, and all other future individuals who become
infected, will each transmit to exactly 5 susceptible individuals every 2 days.
Approach 1.
Approach 2.
Computation by Hand We start by doing some simple
computations by hand to determine whether a pattern
emerges. After 2 days, we have the original 10 people
who have the disease, but we also have 50 more, because
each of the 10 infected individuals has spread the
disease to 5 others. Hence, when t = 2, I = 60. After 2
more days, we have I = 60 + (5 × 60) = 360, and so on.
We could organize our numbers by putting them in a
table, as in Table 1.
Computation via Technology Straightforward iterative
calculations such as those needed for this problem are
easily “programmable” in spreadsheet software such as
Microsoft Excel. Excel is a useful tool for visualizing
step-by-step discretization, and it can allow you to see
the value of each variable at each time step in table
form. If you don’t know how to use Excel, you might
find resources or videos by doing an internet search on
performing iterative calculations in Excel.
If we’ve done our computations in Excel, we can
also create a plot of our solution easily, as in Figure 4.
Table 1. Constant-rate disease model
computations, by hand
0
10
2
60
4
360
6
2160
8
12960
10
77760
12
466560
Figure 4. Graph of constant-rate disease
model output with r=5, τ = 2, and I0 = 10
14000
14000
Infected Population
I, Infected Population
Infected Population
t, In Days 10500
10500
7000
7000
3500
3500
00
00
22
44
Time (days)
Time
(Days)
27
66
88
5: building solutions
Example 2:
Constant-Rate Disease Model (CON’T.)
Approach 3.
Pattern identification We might notice as we perform
computations by hand or by using technology that if
we know the number of infected individuals at some
time step, then we can get the infected population at the
next time step by multiplying by 6. Let’s see why this
happens.
At time t = 0, we have 10 infected individuals. At
time t = τ = 2 days, we have the original 10 infected
individuals plus 50 more, for a total of 60.
For this set of parameters:
For a generic set of parameters:
I(τ) = 10 + 5 ·10
= (1 + 5) 10
= 6 ·10.
I(τ) = I0 + τ I0
= (1 + r) · I0.
For a generic set of parameters:
I(2τ) = 60 + 5 · 60
= (1 + 5) 60
= 6 · 60.
I(2τ) = I(τ) + r · I(τ)
= (1 + r) · I(τ).
For a generic set of parameters:
I(2τ) = 6 · 60
= 6 · (6 · 10)
= 62 · 10.
I(2τ) = (1 + r) I(τ)
= (1 + r) · (1 + r) I0
= (1 + r)2 I0.
You should verify that for I(3τ) we have the following
equations.
For this set of parameters:
I(3τ) = 63 · 10.
For a generic set of parameters:
I(2τ) = (1 + r)3 I0.
You might see a pattern emerging that would lead you
to find a closed form solution for the infected
population after n days have passed.
Continuing, at time t = 2τ = 4 days, we have the 60 from
the previous time step, plus 5 times 60 more.
For this set of parameters:
For this set of parameters:
For this set of parameters:
For a generic set of parameters:
I(nτ) = 6n · 10.
I(nτ) = (1 + r)n I0.
This result, an exponential model, is consistent with
both the values in Table 1 and the corresponding graph
in Figure 4.
The results of each of the three approaches are
perfectly valid model for our stated assumptions, but
we now see that the model may be limited in its ability
to accurately describe some of the real-world characteristics of disease propagation. In the next section we
discuss these questions, and we’ll revisit this model as
a part of model assessment.
What does the formula look like for I(3τ)? (Try it!)
Now we see where multiplication by 6 comes from, but
we might be able to see an even deeper formula emerge.
We can actually substitute our expression for I(τ) into
the equation the equation for I(2τ), as below.
28
Example 3:
varying-Rate Disease Model
(4)
where k is a (positive) proportionality constant. The
larger the value for k, the larger is. So a large k-value
indicates a highly contagious disease.
In order to find a solution to the differential equation, it will be helpful if we can get down to only one
dependent variable. Since we assumed that the population, P, is constant, then we can take advantage of the
relationship P = I(t) + S(t) to write S as S(t) = P − I(t).
Then we can rewrite equation (4) as follows:
= kI(t) (P − I(t)).
Analytic Solution If you’ve studied differential equations
before, you might recognize that this particular
differential equation can be solved analytically using
a technique called separation of variables. (See any
standard calculus text.) Using this technique and
assuming that the initial condition is I(0) = I0, you can
find the solution to be
I(t) =
PI0
I0 + (P − I0 )e -Pkt
.
(6)
Alternately, if you have access a symbolic computation
tool such as Maple, you can use technology to generate
this analytic solution.
With the analytic solution in hand, we can
demonstrate how the model behaves by choosing some
parameter values to generate a plot. (See Figure 5.) Notice
that this model exhibits the same sort of exponential
growth in the initial stages of disease propagation as the
constant-rate disease model, but the rate slows when
there are fewer people left who have not yet contracted
the illness. We see that I approaches 1000 as time
increases. In generating the plot, we used total population
P = 1000 (a parameter), so our model predicts that over
time the entire population contracts the disease.
Figure 5. Analytic solution of the varying-rate disease
model output with k = 0.0006, P = 1000, and I0 = 20
1000
1000
(5)
750
750
Infected Population
= kI(t)S(t),
Approach 1.
Infected Population
We haven’t defined the variables yet in the varyingrate disease model, so let’s do that now and set up the
differential equation which is to be solved.
We define the total population to be P, and each
member of the population must belong to exactly one
of two classes: susceptible, S or infected, I. Hence,
P = S + I. We assume that the total population remains
constant, but the values of S and I change over time, so
we might choose to write P = S(t) + I(t).
Recall that in the varying-rate disease model, we
consider a population in which disease transmission
rate is directly proportional to the product of the number of people infected and the number of people who
are susceptible. We can think of transmission rate as
the rate at which people become infected, or the rate of
change of population I(t). Readers familiar with calculus
may recognize this as the derivative. We will denote this
rate of change I(t) with respect to time as . Then we
can translate the assumption “disease transmission rate
is directly proportional to the product of the number
of people infected and the number of people who are
susceptible” into the equation
500
500
250
250
0
0
0
6.25
0
6.25
12.5
12.5
Time Period (days)
Time period (Days)
29
18.75
18.75
25
25
5: building solutions
Example 3:
varying-Rate Disease Model (CON’t.)
way to predict I(2), and so on. This numerical method,
called the forward Euler method, is easily implemented
in Excel.
Approach 2.
Approximate Solution While we can find an analytic
solution for the differential equation above, many
differential equations, as well as other equations
and systems of equations that describe real-world
phenomena, cannot be solved directly. In these
situations, it is common to approximate a solution, often
referred to as using a “numerical method.” Numerical
methods are a powerful tool in modeling, especially if
you have not yet had formal training in techniques from
calculus and differential equations.
As an example, consider equation (5). Let’s suppose
that we did not have the analytic solution (equation
(6)) to evaluate at any time we choose. One numerical
method would be to try to calculate approximate values
of the infected population at specified later times. We
know the initial population, I(0) = I0, and we have a
model describing the rate of change of the population.
Intuitively we should be able to predict the number of
infected people at a later time, say t = 1. To do this, we
can express the change in the infected population over
that time frame time as
≈
Infected
Population
Infected
Population
Figure 6. Numerical solution of the varying-rate
disease model output with k = 0.0006, P = 1000,
I0 = 20, and time step Δt = 1 day
1000
1000
750
750
500
500
250
250
0
0
0
0
6.25
6.25
12.5
12.5
18.75
18.75
25
25
Time (days)
Time period (Days)
We show a plot of the approximated values of the
infected population at different points in time in Figure
6, and you can see the details of how we obtained these
results in Appendix A.
Notice that the plot of the numerical solution looks
much like the plot of the analytic solution we saw in
Figure 5. While we are pleased that the numerical
solution does a good job of approximating the analytic
solution, we should note that the graphs are not identical. Every numerical method introduces some error, and
the forward Euler method is no exception. The study of
error is a complicated issue, and we will not attempt to
address it here.
I (1) − I (0)
1−0
and plug what we know into the right-hand side of
equation (5). We can then solve for I(1) in terms of I0.
We should be careful, however. What we really have is
an approximation to I(1) because we approximated ,
but our goal was to determine approximate values of
the infected population, and that is what we’ve accomplished. We could then use our value of I(1) in the same
30
summary of building
solutions
Finding a solution to your model may be achieved
by various means, depending on your background
knowledge in mathematics and software. Solving by
ranking is an excellent method for students who do not
have the mathematical training to produce algebraic
formulas. Even with an equation, however, there are
often multiple ways to arrive at a final answer. Software
tools such as Excel can facilitate obtaining solutions.
If you do you use a numerical approximation technique,
it is important to be aware of error that might be
introduced. If nothing else seems to help, try guessand-check.
in summary
1
How you build a solution may depend on
what mathematical tools are available
to you.
2
There is often more than one way to
tackle a problem, so just start and see
what happens.
3
If you don’t immediately know how to
solve the problem at hand, ask yourself
the provided set of questions to help
you get started.
4
Different solution methods can lead to
solutions of different natures. This is
perfectly acceptable.
Activity
Continue working with your model for a ranking
system for roller coasters depending on your
definition of “thrilling.” Use your ranking
system to rank at least 10 roller coasters.
You may find data you need at rcdb.com.
31
6. analysis
AND Model
assessment
We separate this section into two subsections. The first
subsection, Does My Answer Make Sense?, gives some
quick checks to determine whether your solution is at all
reasonable. The second subsection, Model Assessment,
gives more in-depth techniques to analyze the model.
Often we are so excited that we built and solved
a mathematical model that we forget to step back
and carefully examine the results. While this is
understandable since it took hard work to get to that
point, it is essential to ask yourself, “Does my answer
make sense?” Sometimes, the results may indicate a
mistake in the calculations. Other times you may find
that additional or alternate assumptions are needed for
the solution to be realistic. If the results do make sense,
then further analysis is needed to assess the quality of
the model. Recall that open-ended questions may have
more than one solution and that the results depend on
the assumptions made and the level of sophistication
of the mathematics used. An honest evaluation is
necessary to explain when the model is applicable and
when it is not. In this section we will talk about ways
to analyze your results and how to assess the quality of
your model.
32
does my
answer make sense?
During the modeling process, you may gather some
intuition about what the output will be like. Here we
provide some pointers on how to answer the question
“Does my answer make sense?” by analyzing the output
of the model.
• Is the sign of the answer correct? For example, if your
disease model is supposed to calculate the number
of infected people at a certain time, then clearly an
answer of −1000 would not make sense. Carefully
check your calculations, especially if you are using
software. For example, in Excel it is easy to select the
wrong cell when defining a formula. Your model may
be correct, but its implementation may be at fault.
• Is the magnitude of the answer reasonable? If you
are trying to estimate the speed of a car, for example,
then it wouldn’t make sense if your model predicts a
value of 1000 miles per hour. Sometimes, when the
magnitude of a number is off, incorrect units may have
been used somewhere in the process.
• Does the model behave as expected? If the output
of your model is visualized with a graph or plot of
any kind, then carefully look at the intercepts, the
maximum or minimum values, or the long-term
behavior. Were you expecting a horizontal asymptote,
yet your graph just increases without bound? If you
have a data set and believe there is a relationship
between two variables, plot the data. A mathematical
error in the sign of the slope will be immediately
obvious. It could be that you had some assumptions
which were neglected, erroneous units, unrealistic
parameters, or that the software was used incorrectly.
• Can you validate the model? Sometimes it is possible
to validate your model using available data. For
example, if you used your roller coaster ranking model
on the Top Thrill Dragster of Cedar Point, which held
the record for the tallest roller coaster in the world
and goes 120 mph, and the output said it was only
mediocre, then likely your model is not doing what
you want it to.
33
6: analysis and model assessment
model
assessment
Now that you have verified that your model is correct, it
is time to step back and consider the validity of your
model. This includes identifying the strengths and
weaknesses of your model and understanding at a deeper
level the behavior of the model. Performing a sensitivity
analysis, wherein you analyze how changes in the input
and parameters impact the output, can contribute to
understanding the behavior of your model.
software, research other model parameters in order to
fit the model to his or her own needs, or sort through
complicated output to be able to draw a conclusion. It is
also a strength if the people who might use your model
can understand it and have faith in it.
Let’s examine this process by looking at the
constant-rate disease model. Recall our assumption
that each infected person transmits the disease to r
people every τ days. From this, we found the following
exponential function to describe the infected population
after nτ days:
Identifying Strengths and Weaknesses
Once you are convinced that the output is correct and
the model is achieving what you want, assess the quality
of model. This assessment needs be included in the
write-up about your model to help people understand
the conditions under which your model is applicable,
which is strongly linked to the assumptions that were
made along the way. It is necessary for you to provide
an honest, exact assessment of the capabilities of
your model.
This is also a chance to highlight the strengths of the
model. For example, even if a model was formulated
using simple physics, it might require very little expert
knowledge in order to provide meaningful insight.
This can be a huge advantage over a more complex
model that requires the user to program and run
I(nτ) = (1 + r)nI0.
Figure 7 shows the graph of the model output for I0 = 20
infected people, τ = 2 days, and r = 5.
34
Figure 7. Graph of constant-rate disease model output with r = 5, τ = 2, and I0 = 20
Infected
Population
Infected Population
14000
14000
10500
10500
7000
7000
3500
3500
00
00
22
44
66
8
8
Time (Days)
Time (days)
This model has some valuable strengths:
• This model is easy describe to others, which means
that they might have increased confidence in its
output. Allowing full understanding of all parts of the
model can be valuable when trying to understand the
significance of output given different input values.
• This model is not valid for all types of disease. Our
model only has two categories for individuals: susceptible and infected. In real life, after being infected with
some diseases a person recovers and then acquires
immunity to that specific disease. This model wouldn’t
accurately describe that situation.
• We found an analytic solution. If we have values for
I0, τ, and r, the function I(nτ) = (1 + r)nI0 can be used
to solve for the number of infected individuals at
any time.
• This model predicts the disease spreads to everyone.
In other words, under this set of assumptions, the disease will propagate through a population until every
individual has the disease. This (hopefully) is not a
reasonable scenario.
• Our model is consistent with our assumptions. Our
primary assumption is that the disease is spread at
a constant rate. Our model accurately describes
this behavior, and therefore we have developed a
meaningful solution.
This model also has a few notable
weaknesses:
• Our model is quite simple. In addition to being a
strength, it is also a weakness. We may be concerned
with the constant rate assumption, especially since it is
not consistent with our intuition regarding the spread
of disease.
In this case, we have a model based on sound mathematics that provides us with good insight into disease
propagation but also leads to some questionable realworld outcomes if taken as the final answer. In general,
awareness of your model’s capabilities leads to better
overall solutions. In such awareness, you know when it
is appropriate to use your model, and you also have a
starting point from which to build future (more specific
and/or realistic) models. Identifying your model’s weaknesses does not detract from the hard work you’ve done;
it is always preferable for the modeler to acknowledge
weaknesses in the model. If the reader identifies weaknesses that the modeler has missed, then the readers’s
assessment of the model and the modeler suffers.
35
6: analysis and model assessment
sensitivity analysis
Figure 8. Analytic solution of the varying-rate disease model output with P = 1000, I0 = 20, and k ranging from
0.0002 to 0.001 in increments of 0.0002
Infected Population
Infected Population
1000
1000
750
750
500
500
250
250
00
0
0
6.25
6.25
12.5
12.5
18.75
18.75
25
25
Time
(days)
Time
period
(Days)
0.0002
kk= =0.0002
kk ==0.0004
0.0004
k0.0006
= 0.0006
k=
= 0.0008
k =k0.0008
0.001
kk= =
0.0001
does this mean? Well, that depends on the problem you
are solving. In this example, we learned that changes to
k can increase (or decrease) the spread of the disease
to the entire population. If a population were infected
with such a disease, then this model could demonstrate
that a drug that may inhibit the disease transmission
rate could create the time needed to develop an effective
course of treatment.
In our sensitivity analysis, the changes in k were
fairly small (0.0002). How do you know how big (or
small) your variation should be? In some cases you may
have real-world data to help you make a decision. If
that’s not available, use common sense as your guide and
play around with the numbers to develop intuition. For
example, in the variable-rate disease model, we could
also hold k constant and investigate whether changes
to the initial infected population I0 affect our outcome.
In this instance we will definitely vary I0 in increments
larger than 0.0002.
When doing model assessment it is also vital to consider
the model’s sensitivity to changes in the assumptions
and parameters used to build it. A model is considered
sensitive with respect to a parameter if small changes in
that parameter lead to significant changes in the output.
There are several ways to conduct a sensitivity analysis. A simple approach would be to consider a range of
values for a certain parameter while keeping all other
parameters fixed, calculate the output, and then determine the impact on the output. For example let’s take a
look at the effect of changes to the transmission rate in
the variable-rate disease model with P =1000 and I0 =20.
Looking at Figure 8, we see how incremental
changes to the transmission rate affect the output of
our model. In particular, we note that when 0.0006 ≤ k
≤ 0.001, the disease has infected most (if not all) of the
population after 15 days, but it takes just over 20 days
for the disease to reach most of the population when k =
0.0004. When k = 0.0002, a good proportion of the population still has not been infected after 15 days. What
36
model
refinement
flu is over. It doesn’t make sense to put them back into
the susceptible population because we know that they
have developed immunity. So we need to create another
class, R, which represents those who have recovered
from the disease.
We begin by defining the following:
If time allows, assessment and sensitivity analysis
can lead to improvements in the model. Modeling, as
pointed out earlier, is an iterative process, and refinements can almost always be made to develop more
realistic scenarios. If the modeling is being done in a
timed setting, such as for a competition or a homework
assignment, then this may not be possible (although it
is generally possible to indicate the type of refinements
that would improve the model). However, for long-term
projects, the model assessment really is an intermediate
step before (possibly) starting the modeling loop over
again. Discussing possible modifications to the model,
even if you cannot make them, demonstrates that you
are able to think beyond just the first approach. We
demonstrate how this could be done with the variablerate disease model.
In our previous disease model, the population was
split into two classes: infected and not infected. If,
however, we consider something such as an influenza
outbreak, we know that people are capable of transmitting the flu for a short period of time, but eventually
they develop immunity and will no longer spread the
disease. This dynamic is certainly not captured with our
initial disease models. With our new considerations in
mind, we want infected individuals to be able to move
out of the infected population after their bout with the
P = the total number of people in the population,
S = the number of people in the population who are
susceptible,
I = the number of people in the population who are
infected and transmitting the disease,
R = the number of people in the population who have
“recovered” (i.e., they are not susceptible and no
longer transmit the disease).
Notice that for any time, P = S +I +R.
We will assume that each individual who becomes
infected takes the same amount of time to recover. That
is, our model will not take into account the possibility
that one person recovers in 3 days while another takes
5 days to stop being a carrier of the disease (although a
later refinement of our model might include a random
distribution of recovery intervals).
37
6: analysis and model assessment
Figure 9. Graph of constant-rate disease SIR model output for susceptible (S), infected (I), and recovered (R)
populations for parameter values P = 1000, r = 2, τ = 4, and I0 = 12
1000
1000
Population
Infected Population
750
750
500
500
250
250
00
0
6.25
6.25
12.5
12.5
18.75
18.75
25
25
(days)
TimeTime
period
(Days)
S
S
I
I
R
R
as influenza; it starts out slowly, spikes, and then dies
out as the remaining infected population recovers. This
model is not perfect, but it does provide us with insight
into the dynamic interaction of the affected populations.
Let’s move forward looking at a specific example.
Suppose at time t = 0, 12 people of a total population of
1000 become infected with a novel disease. We assume
that each infected person recovers after exactly 4 days
and that each infected person transmits the disease to
2 others during that 4-day period. So we will let τ = 4
days.
At time t = 0, then, we have I0 = 12, S0 = 988, and
R0 = 0. After a 4-day time period, i.e., when t = τ, all 12
of the infected individuals move to recovered status.
In the meantime, they have infected 24 susceptible
individuals. So we have
Table 2. Infected, susceptible, and recovered
populations over time for parameter values P = 1000,
r = 2, τ = 4, and I0 = 12
t
S(t)I(t)R(t)P(t)
0988
1201000
τ 96424121000
I(τ) = 2 · I0 = 2 · 12 = 24,
S(τ) = 988 − I(τ) = 988 − 24 = 964, and
R(τ) = R0 + I0 = 0 + 12.
2τ91648361000
3τ82096841000
4τ 6281921801000
5τ 2443843721000
We can continue as shown in Table 2 and Figure 9.
Our refined model now shows a situation more
consistent with our intuition regarding a disease such
6τ 0 2447561000
38
in summary
1
Activity
Be sure to allocate time to analyze your
Read the solution to the recycling problem.
results since it is indeed a critical part
What are the strengths and limitations of that
of the entire modeling process.
model? What are parameters in the recycling
model that could be examined for sensitivity?
2
Always examine the output you get from
How might the sensitivity of the recycling
your model and ask yourself if it makes
model be analyzed? Write a few paragraphs
sense. If your answer doesn’t make sense,
assessing this model.
verify that you haven’t made a mistake in
implementing your model.
3
If your solution is consistent with your
assumptions but not consistent with the
real-world phenomenon you are trying
to describe, you may need to refine your
model by adjusting your assumptions.
4
5
6
List strengths and weaknessES of
your model.
Try to determine how sensitive your
model is to parameters and assumptions.
Include specific improvements you would
have incorporated given more time.
39
7. putting it all
together
Now that a model has been created, solved, and assessed, the time comes to write everything up as a polished
solution paper. This step is just as important as the effort necessary to get to this point. Keep in mind that you are
the expert about the problem, and now your role is to explain what you did in detail to people unfamiliar with your
solution approach. To this end, it is critical that you take good notes from the initial brainstorming process through
the final analysis to be sure you have kept track of all the assumptions you made. Good writing also takes time, so be
sure to allocate a period of time to step back from the math modeling and focus on quality writing. In this section we
discuss how to structure your report and some key points for successful technical writing.
This step is just as important as the effort
necessary to get to this point. Keep in mind
that you are the expert about the problem.
40
structure
A technical report typically starts with a summary page,
also called an executive summary or an abstract, that is
of one page or shorter. This is not an introduction; it’s
actually a place to summarize how the problem was
solved and to provide a brief description of the results.
It might seem strange to put the conclusion at the
beginning, but this “bottom-line up front” approach is
convenient for those reading your report.
The abstract or summary page should restate the
problem, briefly describe the chosen solution methods,
and provide the final results and conclusions. You
should describe your results in complete sentences
that can stand on its own, without using variables.
The summary lets the reader know what to expect in
the report but does not overwhelm him or her with
unfamiliar mathematical notation. Imagine that a reader
will decide whether to continue reading the rest of your
paper itself based on this abstract. As an example, see
the solution to the recycling problem.
After the summary, the paper should include a
formal introduction that includes a restatement of the
underlying real-world application as if the reader does
not have any prior knowledge. This section usually
contains some motivation or relevant background
information as well but should not include a lengthy
history lesson. Both the general modeling question
as well as the concise problem statement that you
developed should be at the forefront of this section.
This section must provide a paragraph that describes
how you approached the problem. For example, if we
consider the task of ranking roller coasters based on
how thrilling they are, then it would help to define
“thrilling” up front. For example, “Our model is based
on the notion that rides with high accelerations,
inversions, and significant heights are thrilling.”
Note that this statement doesn’t exactly explain how
those features are quantified or implemented for
a mathematical ranking system for thrilling roller
coasters, but it does give the reader an idea of what
will be showing up later in the document. It may seem
counterintuitive, but the introduction and abstract are
typically written last. This is because, after all else has
been written, the author has a complete picture of the
manuscript and may then best tailor these sections
accordingly.
The body of the solution paper will likely be several
pages long and split up into sections about assumptions,
variables, the model, the solution process, analysis,
and overall conclusions. Let the reader know about
the overarching assumptions you made to make the
problem solvable. Some specific assumptions may need
to be included again later, within the paper’s main text,
in order to clarify certain ideas as they are developed.
However, the most important message here is that all
assumptions are included and listed at some point in
your write-up. You should be sure to justify why those
assumptions are reasonable and include citations as
needed. Plagiarism of any kind is never acceptable.
When you next begin to describe your model
and how you solved it, clearly state the variables you
will be using and the corresponding units. If there are
relationships between variables, explain where the come
from and, if needed, refer to any necessary assumptions.
Mathematical equations and formulas should be
centered, each occuring on their own line. We provide
some more specific details on this in the following
section.
Finally, the paper must have a conclusion section
that recaps the important features of the model. It
is critical that this section includes an analysis of
your results, as described in the previous section. An
honest assessment of the strengths and weaknesses is
important. In particular, you can comment on how the
model can be verified and how sensitive the model is to
the assumptions. We proceed by giving some tips about
technical writing.
41
7: putting it all together
TECHNICAL WRITING DOS
& don’ts
1. Do not write a book report.
It is critical that the narrative of your solution doesn’t read like a story about how you came up with your model.
For example, consider the following two excerpts about an assumption made for the recycling model.
Example 1:
A study of drop-off recycling participation in Ohio found that 15.5%
of citizens who do not have access
to curbside recycling use drop-off
recycling [8]. We have assumed that
this data is representative of the U.S.
Example 2:
We were stuck because we did not
know how many people in the U.S.
recycle. We googled and found an
article that Ohio’s participation in
drop-off recycling was 15.5% for
people who did not have access to
curbside recycling, so we used that
number in our model.
The second example is written in
a way that makes the assumption
sound invalid or that it was chosen
only because no other information could be found. However, the
first one sounds as though some
research was done and a useful
and legitimate source was identified, which provided an applicable
statistic.
2. Do not use words when using mathematics would be more appropriate.
Which of the following is more effective?
Example 1:
For an ideal gas, we have
nRT
,
V
where P is the absolute pressure of
the gas, V is the volume of the gas,
n is the amount of substance of gas
(measured in moles), T is the absolute temperature of the gas, and R is
the ideal, or universal, gas constant.
P=
Example 2:
For an ideal gas, the absolute
pressure is directly proportional to
the product of the number of moles
of the gas and the absolute temperature of the gas and inversely proportional to the volume of the gas,
with proportionality constant R, the
ideal, or universal, gas constant.
42
In this case, the mathematics is
easier to follow, and you can
imagine that the more complicated
the calculations, the harder it would
be to try to describe in words.
3. do use proper sentence structure when explaining mathematics.
Communicating mathematics requires proper punctuation, such as periods at the end of a computation if the
computation ends the sentence, as in Example 1 below. Use commas when appropriate.
Example 1:
If s is the length of the side of the square box, then the
area of a side is given by
Example 2:
If s is the length of the side of a square box then we can
find the area and volume.
A = s2,
A = s2
and the volume is given by
V = s3
V = s3.
4. do NOT substitute mathematical symbols for words within sentences, as in the
second of the following two examples.
Example 1:
For this work L is the length of the side of a rectangle.
Example 2:
For this work L = the length of the side of the rectangle.
5. do pay attention to significant figures.
For example, your calculator might read a value of 27.3416927482, but you may not need to report all of those digits
unless you are trying to show accuracy in the later decimal places.
6. do use scientific notation when numbers vary by orders of magnitude,
meaning that the exponent is really what matters in understanding the significance of the value. For example, the
diameter of the sun is 1.391e6 km, while the diameter of a baseball is 2.290e−4 km.
43
7: putting it all together
Technical writing takes practice, but the end result
should be something of which to be extremely proud. In
reviewing your final paper, you can step back and look
at all you have accomplished throughout the modeling
process. Ultimately your model can lead to the creation
of new knowledge and provide deeper insight about the
world we live in. Remember that modeling also takes
practice so the next time you tackle an open-ended
problem you will already have a new set of tools that
will make the entire experience go more smoothly.
7. Do label figures
and use a large enough font so that the axes are
clearly readable.
8. Do not forget to include units as
appropriate.
9. Do check carefully for spelling and
grammar mistakes,
especially those that spell check might miss. For
example, it’s easy to confuse their, there, and they’re.
10. Do give credit where credit is due.
This means including citations and building your
bibliography as you go.
in summary
1
Take notes throughout your entire modeling process so that you do not leave out anything
important, especially assumptions made along the way.
2
Give yourself enough time to focus on the writing process and to proofread the report.
3
Keep in mind that this is a technical document, not a story about your modeling experience.
4
Follow the guidelines for technical writing.
5
Some additional references on technical writing can be found at [3].
6
Pat yourself on the back for your accomplishments.
44
Appendices
& ReferenceS
A. FORWARD EULER
METHOD
Let’s explore how the forward Euler methods works in
the context of the varying-rate disease model.
As a reminder, our model is described with the
differential equation
We start with the initial population, I0 = 20. That
is, when t = 0 days, I = 20. In other words, we know
that the point (0, 20) is on the graph of the solution. We
also know what the slope of the solution curve is at that
point because we have an equation for dIdt :
dI
= kI(t)(P − I(t)),
dt
dI
dt
where I is the number of infected individuals, P is the
total population, and k is a positive constant. We’ll use
the same parameter values we used to plot the analytic
solution: k = 0.0006, P =1000, and I0 =20.
The method we will use here is called the forward
Euler method, which takes advantage of the fact that the
derivative is the same thing as slope of the tangent line.
We’ll demonstrate the method on this example, but do
not want to overwhelm the reader with too many
details. We point you toward [9, 4] for a deeper look
into this very powerful approximation method.
= k I (0) (P − I (0))
t=0
=kI0 (P − I0)
= (0.0006) · 20 · (1000 − 20)
= 11.76.
Now we can write the equation of the line through the
point (0, 20) with slope of 11.76 as follows. Recall that
the independent variable is t and the dependent variable
is I.
45
I − 20 = 11.76(t − 0),
I = 11.76(t − 0) + 20.
appendices & REFERENCES
Figure 10. Numerical solution of the varying-rate disease model output with k = 0.0006, P = 1000, I0 = 20, and
time step Δt = 1 day
Infected Population
Infected Population
1000
1000
750750
500500
250250
0
0
0
0
6.25
6.25
12.5
18.75
12.5
18.75
Time period (Days)
25
25
Time (days)
Hence, we have a line with slope of 11.76 and y-intercept (or I-intercept, in this case) of 20.
We can use this linear function as an approximation for the true solution. We assume that the solution
to the differential equation is approximately the same
as this line for points nearby. Perhaps we’ll assume it’s a
good enough approximation through time t = 1. We can
find the number of infected individuals at t = 1.
dI
dt
= k I (1) (P − I (1))
t=1
= (0.0006) · 31.76 · (1000 − 31.76)
= 18.45.
As before, we can find the equation of the line through
the point (1, 31.76) with slope of 18.45.
I(1) = 11.76(1 − 0) + 20 = 31.76.
I − 31.76 = 18.45(t − 1)
I = 18.45(t − 1) + 31.76.
We will assume that this makes a good enough approximation for the solution through t = 2. Thus we estimate
the population at time t = 2 to be
(Note: While it’s not possible to have a fractional person—or, more specifically, 0.76 of an infected individual
—it doesn’t deter us from continuing to approximate the
solution using this method. We do suggest noting that
something unrealistic has occurred and encourage you
to re-examine it later when assessing your model.) Thus
we have a line segment from (0, 20) to (1, 31.76).
Now you can image us starting the process over. In
other words, we assume that the point (1, 31.76) is on
the solution curve, and we can use the derivative to give
us the slope at that point:
I(2) = 18.45(2 − 1) + 31.76 = 50.21.
Once again, we have identified a solution method
requiring multiple iterative calculations, which can be
easily performed using technology such as Excel.
As before, now that we have a table of values in
Excel, we can generate a plot of our numerical solution.
(See Figure 10.)
46
3
B. the
2013
M
Challenge
Moody’s
Mega
Math
Challenge
2013
Moody’s
Mega
Math
Challenge
2013
Problem and the solution
paper from team 1356
®
®
A contest for high school students
A contest for high school students
SIAM
SIAM
Society for Industrial and Applied Mathematics
Society for Industrial
and
Applied
Mathematics
3600 Market Street, 6th Floor
3600 Market
Philadelphia,
PAStreet,
191046th
USAFloor
Philadelphia,
PA 19104 USA
M3Challenge@siam.org
M3Challenge@siam.org
M3Challenge.siam.org
M3Challenge.siam.org
Waste
WasteNot,
Not,Want
WantNot:
Not:Putting
PuttingRecyclables
RecyclablesininTheir
TheirPlace
Place
Plastics are embedded in a myriad of modern-day products, from pens, cell phones, and storage containers to
Plastics are embedded in a myriad of modern-day products, from pens, cell phones, and storage containers to
car parts, artificial limbs, and medical instruments; unfortunately, there are long-term costs associated with these
car parts, artificial limbs, and medical instruments; unfortunately, there are long-term costs associated with these
advances. Plastics do not biodegrade easily. There is a region of the Northern Pacific Ocean, estimated to be
advances. Plastics do not biodegrade easily. There is a region of the Northern Pacific Ocean, estimated to be
roughly the size of Texas, where plastics collect to form an island and cause serious environmental impact. While
roughly the size of Texas, where plastics collect to form an island and cause serious environmental impact. While
this is an international problem, in the U.S. we also worry about plastics that end up in landfills and may stay
this is an international problem, in the U.S. we also worry about plastics that end up in landfills and may stay
there for hundreds of years. To gain some perspective on the severity of the problem, the first plastic bottle was
there for hundreds of years. To gain some perspective on the severity of the problem, the first plastic bottle was
introduced in 1975 and now, according to some sources, roughly 50 million plastic water bottles end up in U.S.
introduced in 1975 and now, according to some sources, roughly 50 million plastic water bottles end up in U.S.
landfills every day.
landfills every day.
The United States Environmental Protection Agency (EPA) has asked your team to use mathematical modeling to
The United States Environmental Protection Agency (EPA) has asked your team to use mathematical modeling to
investigate this problem.
investigate this problem.
How big is the problem? Create a model for the amount of plastic that ends up in landfills in the United States.
How big is the problem? Create a model for the amount of plastic that ends up in landfills in the United States.
Predict the production rate of plastic waste over time and predict the amount of plastic waste present in landfills
Predict the production rate of plastic waste over time and predict the amount of plastic waste present in landfills
10 years from today.
10 years from today.
Making the right choice on a local scale. Plastics aren’t the only problem. So many of the materials we
Making the right choice on a local scale. Plastics aren’t the only problem. So many of the materials we
dispose of can be recycled. Develop a mathematical model that a city can use to determine which recycling
dispose of can be recycled. Develop a mathematical model that a city can use to determine which recycling
methods it should adopt.You may consider, but are not limited to:
methods it should adopt.You may consider, but are not limited to:
• providing locations where one can drop off pre-sorted recyclables
• providing locations where one can drop off pre-sorted recyclables
• providing single-stream curbside recycling
• providing single-stream curbside recycling
• providing single-stream curbside recycling in addition to having residents pay for each container of garbage
• providing single-stream curbside recycling in addition to having residents pay for each container of garbage
collected
collected
Your model should be developed independent of current recycling practices in the city and should include some
Your model should be developed independent of current recycling practices in the city and should include some
information about the city of interest and some information about the recycling method. Demonstrate how your
information about the city of interest and some information about the recycling method. Demonstrate how your
model works by applying it to each of the following cities: Fargo, North Dakota; Price, Utah; Wichita, Kansas.
model works by applying it to each of the following cities: Fargo, North Dakota; Price, Utah; Wichita, Kansas.
How does this extend to the national scale? Now that you have applied your model to cities of varying
How does this extend to the national scale? Now that you have applied your model to cities of varying
sizes and geographic locations, consider ways that your model can inform the EPA about the feasibility of recycling
sizes and geographic locations, consider ways that your model can inform the EPA about the feasibility of recycling
guidelines and/or standards to govern all states and townships in the U.S. What recommendations does your
guidelines and/or standards to govern all states and townships in the U.S. What recommendations does your
model support? Cite any data used to support your conclusions.
model support? Cite any data used to support your conclusions.
Submit your findings in the form of a report for the EPA.
Submit your findings in the form of a report for the EPA.
The following references may help you get started:
The following references may help you get started:
http://www.epa.gov/epawaste/nonhaz/municipal/index.htm
Moody’s Mega Math Challenge supports
http://www.epa.gov/epawaste/nonhaz/municipal/index.htm
Moody’sof
Mega
Math
Challenge
supports
Mathematics
Planet
Earth
(MPE2013).
http://5gyres.org/what_is_the_issue/the_problem/
Mathematics of Planet Earth (MPE2013).
www.mpe2013.org
http://5gyres.org/what_is_the_issue/the_problem/
www.mpe2013.org
Organized by SIAM
Organized
by SIAM
Society
for Industrial
and Applied Mathematics
Society for Industrial and Applied Mathematics
Funded by
Funded by
47
®
®
Team #1356, Page 1 of 20
Analysis of Plastic Waste Production and Recycling Methods
EXECUTIVE SUMMARY
In 2010 alone, the U.S. generated approximately 250 million tons of trash [1]. Much of
this waste consisted of plastics, which build up in landfills and flow into oceans through storm
drains and watersheds [2], breaking up into little pieces and absorbing contaminants in the
process. A major method to reduce waste is recycling, where materials like glass, paper, and
plastic are reformed to create new products. There are many different methods of collection of
recyclable materials, including drop-off centers, where citizens transport their recyclables; single
stream curbside collection, where the city collects the recyclables of each household; and dual
stream curbside collection, where the city collects recyclables that are pre-sorted by each
household. To encourage or subsidize recycling programs, some cities may implement a Pay-AsYou-Throw (PAYT) program, where citizens pay a fee based on the amount of garbage they
throw away.
The EPA tasked us to analyze the production and discard rate of plastic waste over time.
We were also asked to create a model of possible methods for recycling collection to determine
which methods are appropriate for what cities. Using a linear regression model over years passed
since 2000, we estimated that 35.1 million tons of plastic waste will be discarded in 2023. We
also modeled the use of drop-off centers, single stream curbside collection, and dual stream
curbside collection to calculate the total amount of recyclables collected as well as the cost to the
city using each recycling method.
For collection using drop-off centers, we developed a simulation that randomly
simulated the number of households who would recycle when drop-off centers were placed
around the city. The simulation took into account the area, population, average household,
maximum distance citizens are willing to travel, and number of drop-off centers. Using these
data, we calculated the amount recycled and then calculated the net cost to the city by subtracting
operating costs of Material Recovery Facilities (MRFs) from revenue generated by selling
recycled products.
For curbside collection, we calculated the number of trucks needed to service a given
city, based on population density. Based on labor, upkeep, and fuel, we calculated the costs of a
curbside collection program. Again, using the calculated amount of collected material, we
determined the net revenue generated by these products.
We determined that by using a drop-off center method, Fargo and Wichita would
generate profits, while Price would incur costs that could be partially covered by using Pay-AsYou-Throw. Using a curbside collection method, Fargo and Price would incur costs that could be
partially covered by Pay-As-You-Throw, while Wichita would generate profits using either
single or dual stream collection. Thus, either drop-off or curbside collection methods may be
feasibly implemented in cities around the U.S., depending on population and area of each city.
We concluded that small cities tend to incur net costs from recycling programs, while larger
cities like Wichita may profit from using a dual stream curbside collection program.
To assess use of recycling programs on a national level, we programmed a computer
simulation generating an image of all the counties of the U.S., where blue dots on the U.S. map
represented counties where at least one of our three proposed recycling programs earned a net
profit. In general, we recommend that the EPA extend recycling program guidelines to the
national level.
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Team #1356, Page 2 of 20
I. INTRODUCTION
1. Background
Each year, the U.S. consumes billions of bags and bottles. However, of the plastics that
the U.S. produces, only 5% is recovered [2]. Unrecycled plastics present a growing hazard
because they contain dangerous chemicals like polycarbonate, polystyrene, PETE, LDPE,
HDPE, and polypropylene, which accumulate over time and build up in our oceans and landfills.
As such, it is important to assess the scale of our plastic waste production problem over time.
Our foremost method of reducing wastes like plastics is through recycling, where useful
materials including glass, plastic, paper, and metals are recovered so that they may be used to
create new products [3]. There exist several methods of recycling collection; in general, cities
may use either use drop-off centers or curbside collection. With drop-off centers, the residents
carry the burden of transporting their recyclable waste, while curbside collection places this
burden on the city. If a city implements curbside collection, it may choose to use single stream,
dual stream, or pre-sorted methods; in single stream, all recyclables are collected as one unit,
whereas in dual stream, recyclables are separated into paper and glass, cans, and plastic [4].
Further separation exists with the pre-sorted collection method, where recyclables are fully
separated by material type [5]. There are advantages and disadvantages associated with each
method of collection, and in choosing the type of recycling program to implement, cities must
consider, among other factors, the practicality of individual household collection, as well as the
volume of recyclables that would be collected using each program [6]. Some communities may
use Pay-As-You-Throw (PAYT) programs, which encourage residents to recycle their waste so
as to avoid fees dependent on the weight of their trash [7]. We assess in this analysis whether it is
more efficient to use drop-off centers or curbside collection, depending on the city where the
recycling program is being implemented, as well as the effect of using PAYT programs to
generate additional revenue for the city.
2. Restatement of the Problem
In this analysis, we were requested by the EPA to create a model to predict the change in
plastic production rate over time, as well as the amount of plastic waste in landfills in the year
2023. We were further asked to look at various recycling methods, not limited to the recycling of
plastics, and to analyze the recycling method a city should develop, using as sample points the
cities of Fargo, North Dakota; Price, Utah; and Wichita, Kansas. Finally, the EPA requested that
we provide recommendations for developing recycling methods on the national level based on
the model we designed.
3. Global Assumptions
Throughout our analysis, we will make the following assumptions:
1 A city’s population is approximately evenly distributed. Population mostly varies on a
large scale: in the small microcosm of a city, the population density will not vary much.
2 A city’s shape is approximately square. Most cities are shaped like this, as are the three
sample cities we were provided with.
3 A city’s roads are laid out in a grid plan. The popularity of the grid plan is pervasive,
dating back to Ancient Rome, and most cities are organized as such, like our three sample
cities. [8, 9, 10].
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Team #1356, Page 3 of 20
4 A household’s recycling stance is consistent. That is, a household that recycles always
recycles, and a household that does not will never recycle. Recycling is a habit, and
households that recycle tend to recycle consistently.
II. ANALYSIS OF THE PROBLEM AND THE MODEL
1. Plastic Waste Production
Assumptions
1 We used data collected from the past ten years because the first plastic bottle was
introduced in 1975 [11], and recycling has only become important recently. In other
words, values used before 2000 would not adequately take into account the recycling
methods which have now become widespread.
Approach
We created our model by performing linear and logistic regressions on the amount of
plastic waste discarded per year for the last decade in thousands of tons as provided by the EPA
[1].
Model
Discarded Plastic Waste (thousands of tons) = 463.27 * (years since 2000) + 24443.6
R2 = .803; S = 801
The R2 value of .803 means that 80.3% of the variability in the amount of plastic waste
discarded is explained by the linear relationship between years passed since 2000 and plastic
waste amount. The standard deviation of the residuals was 801.
Based on this model, the amount of unrecycled plastic waste discarded in 2023 will be
463.27 * 23 + 24443.6 = 35098.81 thousands of tons, or 35.1 million tons.
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Team #1356, Page 4 of 20
Discarded Plastic Waste (thousands of tons) = 23733.8 + (28014.1 - 23733.8) / (1 +
exp((Years since 2000 - 2.71325) / -0.752611))
S = 422
The previously mentioned R2 value only makes sense under the assumption that the linear
model was appropriate. Since there is a prominent bend in the data, we fit them with a logistic
curve as well. A statistical software found the four parameters using a successive approximation
method, and produced the model above.
The standard deviation of the residuals in this model is only 422, which is almost twice as
small as it was in the linear model. Unfortunately, this model assumes that the tonnage of
discarded plastic waste will level off, which is not entirely reasonable. It does, however, give a
best-case result (e.g. if recycling initiatives work perfectly). The projected value of discarded
waste for 2023 is 28014.1 thousand tons (within 4 decimal places), which is the maximal value
according to the model.
In summary, the linear model (which seems to overpredict the later values) yields a value
of 35.1 million tons, while the logistic model (which levels off) predicts that it will level off at
28.0 million tons. The US population has been increasing linearly since 2000 [12], so the linear
model gives a more plausible value for the next ten years.
2. Recycling Methods
Assumptions
1 City shape can be approximated as a square or diamond. Most cities in the U.S. are
square-like in shape, including Fargo, North Dakota, Price, Utah, and Wichita, Kansas.
2 The streets of the city are laid out in a grid. Many large cities have streets following a
grid, including Fargo, North Dakota; Price, Utah; and Wichita, Kansas all use grid
systems
3 There is no overlap between use of drop-off centers and curbside collection.
4 The composition of recyclables in the MSW stream is fixed over the entire planning
horizon.
Model 1: Drop-off Centers
Approach
Assumptions
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Team #1356, Page 5 of 20
1 Each household makes a collective decision on whether or not to recycle because it is
convenient for a household to transport all of their recyclables to a drop-off center
together.
2 The probability of a household’s deciding to recycle varies linearly with the household’s
distance to the nearest drop-off center.
3 Recycling households recycle all recyclable waste.
To assess the amount of recyclables collected by a recycling program dependent on dropoff centers, we created a computer simulation where we assumed uniform population density and
where we place equally spread drop-off centers around the city, as many as would fit without
overlapping coverage. To determine whether each household would recycle, a random number
from 0 to 1 is generated, and if the number is less than the household’s probability of recycling,
which we assumed varies linearly with the household’s distance to the nearest drop-off center,
the household recycles. We also determined the cost of maintaining each recycling center and the
revenue the center would generate, and used these data to calculate the total cost to the city of the
drop-off center program. In our simulation, we accepted as inputs the area of the city, population
of the city, average number of people in a household, maximum distance citizens are willing to
travel, and number of drop-off centers.
Taxicab Distance
Because streets are assumed to be organized in a grid, we calculate distance as “taxicab
distance”, or distance in which the only path allowed consists of horizontal and vertical lines. In
other words, given px and py as the coordinates of the drop-off center, and x and y as the
coordinates of the household, the distance between them, d, can be calculated as:
d = |y - py| + |x - px|
A Household’s Maximum Distance to a Drop-off Center
Assumption
1 Recycling households make biweekly trips to a drop-off center.
We recommend that cities conduct a survey to determine the distance their citizens are
willing to travel in order to recycle, though we calculated this distance in our model. U.S.
citizens are willing for their household to pay $2.29 a month for curbside collection [13]. Since
this is the amount that they are willing to pay to recycle at greatest convenience, we can assume
that it is equivalent to the maximum amount they are willing to pay as the driving cost to a dropoff center.
The average price of a gallon of gas is $3.784 [14] and the average mileage of a passenger
car in 2010 was 23.8 mpg [15]. The cost of traveling a distance d is:
Cost = ($3.784/gallon) * d / (23.8 miles/gallon)
The distance citizens are willing to travel each week is:
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Team #1356, Page 6 of 20
($2.29 dollars/month) / (4.35 weeks/month) = $0.53 dollars/week = ($3.784/gallon) * (d
miles/week) / (23.8 miles/gallon)
d miles/week = ($0.53 dollars/week) * (23.8 miles/gallon) / ($3.784 dollars/gallon) =
3.33 miles/week
Since citizens must drive to the drop-off center and back, the maximum distance driven
to the drop-off center is 1.665 miles/week. Assuming that households make biweekly recycling
trips, the maximum distance from a household to a drop-off center for the household to consider
recycling is 1.665 miles/week * 2 weeks = 3.33 miles.
A study of drop-off recycling participation in Ohio supports our model, finding that the
functional usage area of a full-time urban drop-off center is about 3.5 miles [16].
Number of Recycling Households Covered by a Drop-off Center
Assumption
1 The available data from Ohio are representative of the U.S. as a whole.
Each drop-off center will receive recyclables from households up to 3.33 miles away. Using
taxicab distance, which only allows horizontal and vertical movement, the area within 3.33 miles
is bounded by a diamond (a square rotated 45°). The diagonal of the diamond is twice the
distance from the center to a corner, or 2 * 3.33 miles = 6.66 miles. Since the diamond is a
square, diagonal length = square root(2) * side length, so the side length is 4.71 miles. The area
of the diamond is side length ^ 2 = 22.18 sq. mi. This is the coverage area of the drop-off center,
which contains all the households that will consider using the drop-off center.
The number of households in the drop-off center’s coverage area is:
Households = 22.18 sq. mi * (population / land area) / (average household size)
A study of drop-off recycling participation in Ohio found that 15.5% of citizens who do
not have access to curbside recycling use drop-off recycling [16]. Assuming that this data is
representative of the U.S. as a whole, the number of recycling households covered by each dropoff center is:
Recycling households = 22.18 sq. mi * (population / land area) / (average household
size) * .155
In our simulation, we assigned 15.5% as the median household probability of recycling.
The closer a household is to the drop-off center, the more likely it is to recycle. Within the dropoff center coverage area, the closer half of households has a greater than 15.5% recycling
probability and the farther away half of households has a less than 15.5% recycling probability.
The distance from the center to the boundary of the closer half of households is the diagonal of
the square with half the area of the entire coverage area, which is:
Halfway distance = square root(22.18 sq. mi. / 2) * square root(2) = 4.71 miles
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Team #1356, Page 7 of 20
We assumed that the probability of a household recycling varies linearly with the
distance to the nearest recycling center. At a distance of 4.71 miles, the probability is 15.5%. At
the boundary distance of 6.66 miles, the probability is 0%. Extending the line through these
points, at the center, with a distance of 0 miles, the probability is 52.9%. In our simulation, the
number of recycling households covered by each drop-off center is approximately the same as
that calculated using the formula previously given.
Drop-off Center Placement
In our simulation, we placed as many drop-off centers as possible in each city so that
none of the coverage areas overlap, with at least one drop-off center in each city. The cost
efficiency of drop-off centers decreases when their coverage areas overlap.
Annual Amount Recycled
The average American generates 4.5 pounds of waste per day [17], about 75% of which
is recyclable [18]. Thus, the average American generates 4.5 pounds * 0.75 = 3.375 pounds of
recyclable waste per day.
Annual Amount Recycled (tons) = (recycling households) * (average household size) *
3.375 lb * 365 days/year * 0.005 lb/ton * (# drop-off centers)
This formula can be used in place of our simulation to calculate annual amount recycled,
as long as there is no overlap between drop-off center coverage areas and the drop-off center
coverage area is entirely contained within the city. For example, because the drop-off center
coverage area (22.18 sq. mi.) is much larger than the area of Price, Utah (4.2 sq. mi.), this
formula cannot be used in place of our simulation for Price, Utah.
Using our simulation, we were able to calculate the annual amount recycled for Fargo,
North Dakota; Price, Utah; and Wichita, Kansas, as well as to visualize the households
contributing recyclables to each city. In the screenshots below, the white dots represent the dropoff centers; the green dots represent households that are recycling; and the red dots represent
households that are not recycling.
Fargo, North Dakota
Annual Amount Recycled (tons) = 5209.66
Number of people recycling= 8458
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Team #1356, Page 8 of 20
Land Area = 48.82 sq. mi.
Population = 105,549 people
Average household size = 2.15 people
Price, Utah
Annual Amount Recycled (tons) = 1876.88
Number of people who recycle = 3047
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Team #1356, Page 9 of 20
Land Area = 4.2 sq. mi.
Population = 8,402 people
Average household size = 2.60 people
Wichita, Kansas
Annual Amount Recycled (tons) = 16929.56
Number of people who recycle = 27486
Land Area = 159.29 sq. mi.
Population = 382,368 people
Average household size = 2.48 people
Drop-off Center Cost
A report by design engineering company R.W. Beck, Inc. recommends front load
dumpsters as the most cost-effective type of drop-off center. Under this plan, front load
dumpsters would be set up at each drop-off center site and recyclables would be collected in two
streams, commingled containers and paper. The annual cost of a front load dumpster site is about
$5,575 per year [19]. Thus, the total annual cost of drop-off centers is:
Annual cost of drop-off centers = $5,575 * (# drop-off centers)
Revenue Generated
To calculate the total revenue per ton generated from selling recycled products, we used
the following formula, taking into account the market price per ton for each product [20, 21, 22,
23, 24, 25, 26, 27]:
Revenue per ton = RevenuePaper + RevenueGlass + RevenueFerrous Metals + RevenueAluminum +
RevenuePlastic + RevenueTextiles + RevenueWood = (.7012*$112.82) + (.0492*$13) +
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Team #1356, Page 10 of 20
(.1131*$217.75) + (.0107*$310) + (.0401*$370) + (.031*$100) + (.0362*$296)
+($135*.0180) = $128.78 per ton of recycled material
[1]
Based on a study conducted on recycling collection and processing options in New
Hampshire [28], cities can decide between small, medium, and large Materials Recovery
Facilities (MRFs) depending on the annual tonnage. The cost per ton using dual stream and cost
per ton using single stream varies depending on the size of the MRF. For drop-off centers, we are
assuming that dual stream is used.
Fargo, North Dakota
We calculated that Fargo would collect 5,209.66 tons of recyclables. This suggests that a
medium tonnage mini MRF, which has an annual tonnage of 5,283, is sufficient for the city. The
cost per ton of a medium mini MRF using dual stream is $124.62. Since the material revenue per
ton was previously found to be $128.78, we can calculate the net cost per ton as:
Net cost = $124.62 - $128.78 = -$4.16
The total cost to the city can then be calculated as:
Total cost = -$4.16 per ton * 5,209.66 tons + $5,575 per drop-off container * 1 container
= -$16,097.19 (profit)
Price, Utah
We calculated that Price would collect 1876.88 tons of recyclables. Price would use a low
tonnage mini MRF, and the net cost per ton would also be $89.69. Then, the total cost to the city
is:
Total cost = $89.69 per ton * 1876.88 tons + $5,575 per drop-off container * 1 container
= $173,912.37
Wichita, Kansas
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Team #1356, Page 11 of 20
We calculated that Wichita would collect 16,929.56 tons of recyclables, suggesting that
Wichita would require a high tonnage mini MRF, which has an annual tonnage of around 7,500.
For a high tonnage mini MRF, the cost per ton for dual stream is $95.40. Since the material
revenue is $128.78 per ton, the net cost per ton is:
Net cost = $95.40 - 128.78 = -33.38
This represents a profit of $33.38 per ton of recycled material. The total cost to the city is then:
Total cost = -$33.38 per ton * 16,929.56 tons + $5,575 per drop-off container * 1
container = -$559,533.71 (profit)
Pay-As-You-Throw
If the city implements a Pay-As-You-Throw (PAYT) program, it will collect revenue
from citizens who must pay an amount depending on the volume of waste they generate. We can
calculate revenue generated by such a program by using the formula [29]:
RevenuePAYT = (WeightWaste / VolumeContainer * PriceContainer - PriceStartup, Maintenance per day) *
Population
The average American generates 4.5 lbs of waste and recycles 1.5 lbs [17]. PAYT
programs cost around $0.28 per capita, based on surveys of Wisconsin and Iowa [30]. We also
simplified VolumeContainer * PriceContainer as Container price/pound, since the containers are meant
to hold specific amounts of weight. Thus, the revenue generated by PAYT for citizens who
recycle can be calculated as:
RevenuePAYT, Recycle = ((4.5 lbs - 3.375 lbs) * Container price/pound - $0.28/365) *
PopulationRecycle
RevenuePAYT, Don’t recycle = (4.5 lbs * Container price/pound - $0.28/365) * PopulationDon’t recycle
Total revenuePAYT = RevenuePAYT, Recycle + RevenuePAYT, Don’t recycle = ((4.5 lbs - 3.375 lbs) *
Container price/pound - $0.28/365) * PopulationRecycle+ (4.5 lbs * Container price/pound $0.28/365) * PopulationDon’t recycle
Fargo, North Dakota
Total revenuePAYT = (1.125 lbs * x - $0.28/365) * 8,458 peopleRecycle + (4.5 lbs * x - $0.28/365) *
(105,549 - 8,458 peopleDon’t recycle)
Price, Utah
Total revenuePAYT = (1.125 lbs * x * - $0.28/365) * 3047 peopleRecycle + (4.5 lbs * x - $0.28/365)
* (8,402 - 3,047 peopleDon’t recycle)
Wichita, Kansas
Total revenuePAYT = (1.125 lbs * x - $0.28/365) * 27,486 peopleRecycle + (4.5 lbs * x - $0.28/365)
* (382,368 - 27,486 peopleDon’t recycle)
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Team #1356, Page 12 of 20
The following table provides the total revenue generated by a PAYT program if the
container price per pound were $0.01, $0.05, or $0.10.
Sensitivity Analysis
We tested the sensitivity of our simulation of the annual amount recycled in a city using a
drop-off recycling program. We changed population and area by +/- 2%, 5%, and 10% and
examined the resulting change in annual amount recycled. For simplicity, we only examined the
changes for one of our sample cities: Fargo, North Dakota.
The annual amount recycled responds approximately linearly to both area and population.
The response is not precisely linear because the randomness used in the simulation to determine
whether each household recycles introduces some variation between different runs of the
simulation. Since the slopes are small, a slight error in the initial parameters would not
significantly change the simulation’s output.
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Team #1356, Page 13 of 20
Model 2: Curbside Collection
Assumptions
1 Each city has only one recycling processing plant, located at the geographic center, as we
found that one large-scale processing center is more than enough to cover one city’s
recycling needs.
2 Recycling collection comes biweekly.
Approach
We subdivided the city into zones for which one garbage truck was responsible. Each
truck is responsible for driving to its zone, collecting all the recyclable waste it can, and
delivering it to the central processing center, which then sorts and processes the recyclable waste.
Recyclable Waste Collected and Cost to City
We divide the cost to the city into three parts: the cost of gasoline, the wages of the truck
drivers, and the price of truck upkeep. The cost is as follows:
Cost = (Price of diesel fuel in dollars/gallon) * distance / (Truck miles/gallon) + (Num houses) /
(Houses/hour) * (Driver wage/hour) + Truck_Upkeep
The number of houses visited per hour varies depending on whether a single stream or
dual stream collection method is used; for single stream, 171 households are visited per hour,
while for dual stream, 130 households are visited per hour [31]. The mileage of a truck is 5 mpg,
with a cost of $4.02 per gallon. The average wage of a truck driver is $16 dollars/hour [19].
Thus, the formulas for single stream and dual stream collection costs are as follows:
Single stream cost = ($4.02 dollars/gallon) * distance / (5 miles/gallon) + (Num houses) / (171
houses/hour) * ($16 wage/hour)
Dual stream cost = ($4.02 dollars/gallon) * distance / (5 miles/gallon) + (Num houses) / (130
houses/hour) * ($16 wage/hour)
We assume that a truck driver can only collect for 7 hours a day: (8 hour work day, minus
an hour for lunch and driving). So, a truck driver has a maximum amount of households s/he can
visit in a biweekly circuit (171 * 7 * 10=11970 for single-stream and 130*7*10=9100 for dualstream). When a driver is tasked with more houses that s/he can visit, we simply used this
ceiling. To demonstrate, the graphs below show efficiency, in terms of tons of recyclable waste
collected per thousand dollars, versus the number of trucks in each city.
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Team #1356, Page 14 of 20
Using the model, we calculated the optimum number of trucks for each city for either
dual stream or single stream curbside collection depending on the efficiency of the collection,
quantified using the tons of recycled material collected per $1,000, and the total amount of
recycled waste collected. The results for optimum number of trucks are shown below:
City
Single Stream
Dual Stream
Fargo
5
6
Price
1
1
Wichita
13
17
Given the optimum number of collection trucks, the annual cost and tons of recyclable
waste collected can then be determined using our computer simulation.
City
Single Stream
Dual Stream
Tons of Waste
Collection Cost
Tons of Waste
Collection Cost
Fargo
25292.85
$205,787.20
25933.39
$319,383.30
Price
2064.37
$24,526.54
2064.37
$27,005.94
Wichita
93947.82
$713,424.24
88719.32
$798,496.11
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Team #1356, Page 15 of 20
To calculate the revenues and costs generated or incurred from curbside collection, we
needed to determine the cost of recycling and sorting at large-scale MRFs. To calculate the net
costs per ton of material in processed in a MRF, we used data from Resource Recycling Systems
[31] to find operating, capital, and maintenance costs for MRFs of different tonnage capacity. A
graphical representation of the processing and operating costs is shown below[31]:
Fargo, North Dakota
Single stream:
Using our model, we calculated that Fargo would generate 25,292.85 tons of recyclables
annually using single stream curbside pickup. The operating cost is about $130 per ton for a dual
stream MRF of the same tonnage capacity [31]. However, single stream MRFs have greater
processing costs in the range of $10-15 per ton (averaged at $12.5), because of greater sorting
required [4]. Using the revenue generated from selling recovered material, as calculated in the
Drop-Off Center section to be $128.78, the net cost and total cost are:
Net cost per ton = ($130+ $12.5 )- $128.78 + = $13.72 per ton
Total cost = 25,292.85 tons * $1.22 per ton + Collection cost = $347,017.90 +
$205,787.20 = $552,805.1
Dual stream:
Using our model, we calculated that Fargo would generate 25,933.39 tons of recyclables
annually using dual stream. The operating cost is about $130 per ton. Thus, net cost and total
cost are:
Net cost per ton = $130 - 128.78 = $1.22 per ton
Total cost = 25,933.39 tons * $1.22 per ton + Collection cost = $31,638.74 +
$319,383.30 = $351,022.04
Price, Utah
Single stream:
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Using our model, we calculated that Price would generate 2,064.37 tons of recyclables
annually using single stream. A mini MRF, with an annual tonnage of 2,649, is sufficient. The
operating cost is about $245.62 per ton for single stream. Thus, net cost and total cost are:
Net cost per ton = $245.62 - 128.78 = $116.84
Total cost = 2,064.37 tons * $116.84 per ton + Collection cost = $24,526.54 + $241,201
= $265,727.53
Dual stream:
Using our model, we calculated that Price would generate 2,064.37 tons of recyclables
annually using dual stream. A mini MRFis again sufficient. The operating cost is about $218.47
per ton for dual stream. Thus, net cost and total cost are:
Net cost per ton = $218.47 - 128.78=89.69
Total cost = 2064.37 tons * $89.69 per ton + Collection cost = $24,526.54 + $27,005.94
= $212,159.29
Wichita, Kansas
Single stream:
Using our model, we calculated that Wichita would generate 93,947.82 tons of
recyclables annually using single stream. Thus, net cost and total cost are:
Net cost per ton = ($95 + $12.5) - $128.78 = -$21.28
Total cost = 93,947.82 tons * -$21.28 per ton + Collection cost = -$3,173,557 +
$713,424.24 = -$1,285,785.61
As the cost is negative, the city receives a profit.
Dual stream:
Using our model, we calculated that Wichita would generate 88,719.32 tons of
recyclables annually using dual stream. Thus, net cost and total cost are:
Net cost per ton = $95 - $128.78 = -$33.78
Total cost = 93,947.82 tons * -$33.78 per ton + Collection cost = -$3,173,557.36 +
$798,496.11 = -$2,375,061
The city again receives a profit.
Pay-As-You-Throw
We can apply the Pay-As-You-Throw revenue formulas calculated in the Drop-Off
Centers section:
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Total revenuePAYT = RevenuePAYT, Recycle + RevenuePAYT, Don’t recycle = ((4.5 lbs - 3.375 lbs)
* Container price/pound - $0.28/365) * PopulationRecycle+ (4.5 lbs * Container price/pound $0.28/365) * PopulationDon’t recycle
Given that 40% of people to whom curbside recycling is available recycle [16], we
calculated the total revenue each city can expect from a Pay-As-You-Throw program alongside
curbside recycling. The variable “x” is used to represent the container price per pound, which is
up to the city to set.
Fargo, North Dakota
Total revenuePAYT = (1.125 lbs * x - $0.28/365) * (.40 * 105,549 peopleRecycle) + (4.5 lbs * x $0.28/365) * (105,549 - .40 * 105,549 peopleDon’t recycle)
Price, Utah
Total revenuePAYT = (1.125 lbs * x - $0.28/365) * (0.40 * 8,402 peopleRecycle) + (4.5 lbs * x $0.28/365) * (8,402 - 0.40 * 8,402 peopleDon’t recycle)
Wichita, Kansas
Total revenuePAYT = (1.125 lbs * x - $0.28/365) * (0.40 * 382,368 peopleRecycle) + (4.5 lbs * x $0.28/365) * (382,368 - 0.40 * 382,368 peopleDon’t recycle)
The following table provides the total revenue generated by a PAYT program if the
container price per pound were $0.01, $0.05, or $0.10.
3. Testing the Models
To test our models for accuracy, cities with a current drop-off recycling, single-stream
curbside collection, or dual-stream curbside collection program can be run through the models.
The population, area, and other required attributes of the city will be input into our models, and
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Team #1356, Page 18 of 20
the accuracy of our models would be confirmed if the model results for annual amount recycled
and net annual cost to the city are similar to the values in reality.
4. Recommendations
When designing a recycling program, the city should identify markets for recycled
materials. The characteristics of the market determine how recyclables should be collected,
processed, and eventually sold [32].
To extend our model to the national level, we took US Census data from 2000 which
recorded the population density. We then tested each county. On the national level, the EPA
should strongly encourage recycling programs for almost every county or region, particularly in
denser, less rural regions.
The diagram below marks all the centers of all the counties where at least one of our
three proposed recycling programs turns a profit to the community, based on the models were
proposed earlier.
In general, across the U.S., very small cities such as Price, Utah will incur losses from a
recycling program. A drop-off program cannot be used to full advantage because much of the
potential coverage area of one drop-off center lies beyond the city limits. In relatively large,
densely populated cities such as Wichita, Kansas, dual-stream curbside collection is generally
recommended to bring the highest profits. This is due to low participation in drop-off recycling;
on average, only 15.5% of potentially covered households participate. The revenue benefits of a
pay-as-you-throw initiative must be balanced against the cost of its unpopularity among citizens.
A pay-as-you-throw initiative is generally recommended for small cities such as Price, Utah that
seek to adopt a recycling program but incur losses no matter what the program. In these cases, a
pay-as-you-throw initiative is recommended to offset losses to the city.
III. CONCLUSION
Effective recycling programs are critical for cities to address the waste accumulation in
landfills. Based on our models, we conclude that drop-off centers, curbside collection, and payas-you-throw initiatives can all be feasible recycling programs, depending on the population and
area of a given city. All the models are resistant to minor changes in the input values and can be
applied to any city.
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Team #1356, Page 19 of 20
The population growth of the U.S. has a notable effect on the change in the amount of
plastic waste discarded in landfills each year. Partly because U.S. population growth has been
linear in recent years, we determined that a linear model was most appropriate for predicting the
amount of plastic waste discarded. Our linear model predicts that 35.1 million tons of plastic
waste will be discarded in 2023, an increase of 13% over 2010.
Using a drop-off program, Fargo, North Dakota, and Wichita, Kansas would both
generate profits from the sales of recovered materials. The net profits leave an unpopular PAYT
initiative unnecessary. In Price, Utah, however, a drop-off program sustains losses because of the
very small size of the city. For this reason, we recommend that Price adopt a PAYT initiative to
raise revenues and offset costs of the drop-off program.
Using any curbside program, single-stream or dual-stream, Fargo, North Dakota and
Price, Utah incur losses. In both cities, a drop-off program is recommended: in Fargo, because a
drop-off generates profits and in Price, a drop-off generates less losses than a curbside collection.
In Wichita, Kansas, however, both a single and dual stream curbside collection generate a profit,
leaving all three programs feasible. Dual-stream curbside collection is strongly recommended,
however, for the highest profits.
Nationally, small cities generally incur losses with any recycling program, as seen in our
model results for Price, Utah. Dual-stream curbside collection is generally recommended for
large, densely populated cities, who can take advantage of efficiencies of scale. The revenue
benefits of a PAYT initiative must be balanced against the cost of its unpopularity among
citizens, though it is recommended for small cities to help offset their losses. Recycling has
environmental benefits for any city but is especially important for large, densely populated cities,
where it has economic as well as environmental benefits.
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68
motivation and acknowledgments
We came to this project as members of the Moody’s Mega Math Challenge’s Problem Development Committee,
where we have responsibility for soliciting, writing, editing, and vetting potential problems to use in this math
modeling scholarship contest. Something that was made clear from the start was that many high school students are
unfamiliar with how to do modeling, how to get started, and how to reach a solution. In addition, we have taken part
in NSF workshops where the focus is on engaging math education, motivating students to study and pursue careers
in STEM, and keeping them in those disciplines when the going gets rough through better understanding and
enthusiasm for the subjects.
The support of several individuals and organizations was critical to making the handbook happen: First, Frances
G. Laserson, President of The Moody’s Foundation, which has funded the Moody’s Mega Math Challenge since its
inception in 2006. Her commitment to education, specifically in applied math, economics, and finance, has been
instrumental in enabling us work on the content contained here. Dr. Peter Turner, Dean of Arts and Sciences at
Clarkson University and Vice President for Education and Chair of the Education Committee for the Society for
Industrial and Applied Mathematics (SIAM), has supported both the Challenge and SIAM’s increased attention to the
high school and undergraduate communities of students interested in STEM and, in particular, math modeling. Peter
has also been principal investigator, along with Dr. James C. Crowley, executive director of SIAM, on the “Modeling
across the Curriculum” workshops funded by the National Science Foundation (NSF) and through which the content
of this handbook was also enriched. And finally, Michelle Montgomery, SIAM Director of Marketing and Outreach
and Project Director for Moody’s Mega Math Challenge, and her staff team at SIAM. Michelle’s group organizes the
Moody’s Mega Math Challenge, participates enthusiastically in the NSF workshops, and seeks to optimize outreach
to young people through multiple projects and channels to fulfill one of SIAM’s key goals: increasing the pipeline of
individuals going into applied mathematics and computational science programs and careers.
#WeDidIt
Karen, Katie and Ben
this handbook was made possible with enthusiasm and funding from the following organizations:
National Science Foundation
3600 Market Street, 6th Floor
Philadelphia, PA 19104-2688 USA
www.siam.org